ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

Micron Document

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<title>Moran process</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Moran process</span></span>

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<p>A <b>Moran process</b> or <b>Moran model</b> is a simple <a href="Stochastic_process" title="Stochastic process">stochastic process</a> used in <a href="Biology" title="Biology">biology</a> to describe <a href="https://en.wiktionary.org/wiki/finite" class="extiw external" title="wikt:finite">finite</a> populations. The process is named after <a href="Pat_Moran_(statistician)" class="mw-redirect" title="Pat Moran (statistician)">Patrick Moran</a>, who first proposed the model in 1958.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It can be used to model variety-increasing processes such as <a href="Mutation" title="Mutation">mutation</a> as well as variety-reducing effects such as <a href="Genetic_drift" title="Genetic drift">genetic drift</a> and <a href="Natural_selection" title="Natural selection">natural selection</a>. The process can describe the probabilistic dynamics in a finite population of constant size <i>N</i> in which two <a href="Allele" title="Allele">alleles</a> A and B are competing for dominance. The two alleles are considered to be true <a href="Replicator_(evolution_unit)" class="mw-redirect" title="Replicator (evolution unit)">replicators</a> (i.e. entities that make copies of themselves).

</p><p>In each time step a random individual (which is of either type A or B) is chosen for reproduction and a random individual is chosen for death; thus ensuring that the population size remains constant. To model selection, one type has to have a higher fitness and is thus more likely to be chosen for reproduction.

The same individual can be chosen for death and for reproduction in the same step.

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<div class="mw-heading mw-heading2"><h2 id="Neutral_drift">Neutral drift</h2></div>

<p>Neutral drift is the idea that a <a href="Neutral_mutation" title="Neutral mutation">neutral mutation</a> can spread throughout a population, so that eventually the original <a href="Allele" title="Allele">allele</a> is lost. A neutral mutation does not bring any <a href="Fitness_(biology)" title="Fitness (biology)">fitness</a> advantage or disadvantage to its bearer. The simple case of the Moran process can describe this phenomenon.

</p><p>The Moran process is defined on the state space <span class="texhtml"><i>i</i> = 0, ..., <i>N</i></span> which count the number of A individuals. Since the number of A individuals can change at most by one at each time step, a transition exists only between state <i>i</i> and state <span class="texhtml"><i>i</i> − 1, <i>i</i></span> and <span class="texhtml"><i>i</i> + 1</span>. Thus the <a href="Stochastic_matrix" title="Stochastic matrix">transition matrix</a> of the stochastic process is <a href="Tridiagonal_matrix" title="Tridiagonal matrix">tri-diagonal</a> in shape and the transition probabilities are

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P_{i,i-1}&={\frac {N-i}{N}}{\frac {i}{N}}\\P_{i,i}&=1-P_{i,i-1}-P_{i,i+1}\\P_{i,i+1}&={\frac {i}{N}}{\frac {N-i}{N}}\\\end{aligned}}}</annotation>

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</math></span><img src="./64bbbd06e3de01525aa3444260652efacfe3e513.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.176ex; margin-bottom: -0.328ex; width:26.945ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}P_{i,i-1}&={\frac {N-i}{N}}{\frac {i}{N}}\\P_{i,i}&=1-P_{i,i-1}-P_{i,i+1}\\P_{i,i+1}&={\frac {i}{N}}{\frac {N-i}{N}}\\\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>The entry <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i,j}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{i,j}}</annotation>

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</math></span><img src="./a423e710a1203538cb2771f803dc8a8c74bb44a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.427ex; height:2.843ex;" alt="{\displaystyle P_{i,j}}" loading="lazy"></span> denotes the probability to go from state <i>i</i> to state <i>j</i>. To understand the formulas for the transition probabilities one has to look at the definition of the process which states that always one individual will be chosen for reproduction and one is chosen for death. Once the A individuals have died out, they will never be reintroduced into the population since the process does not model <a href="Mutation" title="Mutation">mutations</a> (A cannot be reintroduced into the population once it has died out and <i>vice versa</i>) and thus <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0,0}=1}">

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<annotation encoding="application/x-tex">{\displaystyle P_{0,0}=1}</annotation>

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</math></span><img src="./9a8679c4b4b81708bb52bd779e39f0313f0cf493.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.087ex; height:2.843ex;" alt="{\displaystyle P_{0,0}=1}" loading="lazy"></span>. For the same reason the population of A individuals will always stay <i>N</i> once they have reached that number and taken over the population and thus <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{N,N}=1}">

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<annotation encoding="application/x-tex">{\displaystyle P_{N,N}=1}</annotation>

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</math></span><img src="./850c3e28096bc4ff35647ee3b63d64aaffee2db9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.361ex; height:2.843ex;" alt="{\displaystyle P_{N,N}=1}" loading="lazy"></span>. The states 0 and <i>N</i> are called <i>absorbing</i> while the states <span class="texhtml">1, ..., <i>N</i> − 1</span> are called <i>transient</i>. The intermediate transition probabilities can be explained by considering the first term to be the probability to choose the individual whose abundance will increase by one and the second term the probability to choose the other type for death. Obviously, if the same type is chosen for reproduction and for death, then the abundance of one type does not change.

</p><p>Eventually the population will reach one of the absorbing states and then stay there forever. In the transient states, random fluctuations will occur but eventually the population of A will either go extinct or reach fixation. This is one of the most important differences to deterministic processes which cannot model random events. The <a href="Expected_value" title="Expected value">expected value</a> and the <a href="Variance" title="Variance">variance</a> of the number of A individuals <span class="texhtml"><i>X</i>(<i>t</i>)</span> at timepoint <i>t</i> can be computed when an initial state <span class="texhtml"><i>X</i>(0) = <i>i</i></span> is given:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} [X(t)\mid X(0)=i]&=i\\\operatorname {Var} (X(t)\mid X(0)=i)&={\tfrac {2i}{N}}\left(1-{\tfrac {i}{N}}\right){\frac {1-\left(1-{\frac {2}{N^{2}}}\right)^{t}}{\frac {2}{N^{2}}}}\end{aligned}}}</annotation>

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</math></span><img src="./7777883b1fe286e05272451c0807bfa638461e16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.653ex; margin-bottom: -0.185ex; width:52.651ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {E} [X(t)\mid X(0)=i]&=i\\\operatorname {Var} (X(t)\mid X(0)=i)&={\tfrac {2i}{N}}\left(1-{\tfrac {i}{N}}\right){\frac {1-\left(1-{\frac {2}{N^{2}}}\right)^{t}}{\frac {2}{N^{2}}}}\end{aligned}}}" loading="lazy"></span></dd></dl>

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</style><div class="hidden-begin mw-collapsible mw-collapsed" style=""><div class="hidden-title skin-nightmode-reset-color" style="text-align:center; background-color: #f2dfce; color:black;">For a mathematical derivation of the equation above, click on "show" to reveal</div><div class="hidden-content mw-collapsible-content" style="border:1px #C4C3D0 solid;">

<p>For the expected value the calculation runs as follows. Writing <span class="texhtml"><i>p</i> = <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>i</i></span><span class="sr-only">/</span><span class="den"><i>N</i></span></span>⁠</span>,</span>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} [X(t)\mid X(t-1)=i]&=(i-1)P_{i,i-1}+iP_{i,i}+(i+1)P_{i,i+1}\\&=2ip(1-p)+i(p^{2}+(1-p)^{2})\\&=i.\end{aligned}}}</annotation>

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</math></span><img src="./d0c27d92ed8b2ec8dc4beaa792342b43dd84fc81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:59.689ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {E} [X(t)\mid X(t-1)=i]&=(i-1)P_{i,i-1}+iP_{i,i}+(i+1)P_{i,i+1}\\&=2ip(1-p)+i(p^{2}+(1-p)^{2})\\&=i.\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>Writing <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=X(t)}">

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</math></span><img src="./740e9a0009e0bcedce98f70d0a0fe95666b69c4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.501ex; height:2.843ex;" alt="{\displaystyle Y=X(t)}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=X(t-1)}">

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<annotation encoding="application/x-tex">{\displaystyle Z=X(t-1)}</annotation>

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</math></span><img src="./c2a53038bde86c931376befdb7e956ba7fab70df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.411ex; height:2.843ex;" alt="{\displaystyle Z=X(t-1)}" loading="lazy"></span>, and applying the <a href="Law_of_total_expectation" title="Law of total expectation">law of total expectation</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [Y]=\operatorname {E} [\operatorname {E} [Y\mid Z]]=\operatorname {E} [Z].}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [Y]=\operatorname {E} [\operatorname {E} [Y\mid Z]]=\operatorname {E} [Z].}</annotation>

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</math></span><img src="./ce5a1cf2916d2323a7d1ce652548499e0e692746.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.194ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [Y]=\operatorname {E} [\operatorname {E} [Y\mid Z]]=\operatorname {E} [Z].}" loading="lazy"></span> Applying the argument repeatedly gives <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X(t)]=\operatorname {E} [X(0)],}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X(t)]=\operatorname {E} [X(0)],}</annotation>

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</math></span><img src="./9d5fcf30d8da3104d140577ab9d1e373e2dbb6ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.079ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [X(t)]=\operatorname {E} [X(0)],}" loading="lazy"></span> or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X(t)\mid X(0)=i]=i.}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X(t)\mid X(0)=i]=i.}</annotation>

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</math></span><img src="./eaf582aba43edd36be33dc39a41d9d3abcbaf3fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.843ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [X(t)\mid X(0)=i]=i.}" loading="lazy"></span>

</p><p>For the variance the calculation runs as follows. Writing <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{t}=\operatorname {Var} (X(t)\mid X(0)=i),}">

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<annotation encoding="application/x-tex">{\displaystyle V_{t}=\operatorname {Var} (X(t)\mid X(0)=i),}</annotation>

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</math></span><img src="./f2a8a1f47716e80748cf6cb806fb93764a217ec0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.972ex; height:2.843ex;" alt="{\displaystyle V_{t}=\operatorname {Var} (X(t)\mid X(0)=i),}" loading="lazy"></span> we have

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V_{1}&=E\left[X(1)^{2}\mid X(0)=i\right]-\operatorname {E} [X(1)\mid X(0)=i]^{2}\\&=(i-1)^{2}p(1-p)+i^{2}\left(p^{2}+(1-p)^{2}\right)+(i+1)^{2}p(1-p)-i^{2}\\&=2p(1-p)\end{aligned}}}</annotation>

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</math></span><img src="./690876fb84dd1b4e676475fb2e8fbdbb61867afb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:65.75ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}V_{1}&=E\left[X(1)^{2}\mid X(0)=i\right]-\operatorname {E} [X(1)\mid X(0)=i]^{2}\\&=(i-1)^{2}p(1-p)+i^{2}\left(p^{2}+(1-p)^{2}\right)+(i+1)^{2}p(1-p)-i^{2}\\&=2p(1-p)\end{aligned}}}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle (X(t)\mid X(t-1)=i)}</annotation>

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</math></span><img src="./62f71e76a66978ee00d0aa3831a8dafca293aaff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.908ex; height:2.843ex;" alt="{\displaystyle (X(t)\mid X(t-1)=i)}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X(1)\mid X(0)=i)}">

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<annotation encoding="application/x-tex">{\displaystyle (X(1)\mid X(0)=i)}</annotation>

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</math></span><img src="./a7611a5401668a9def202da15c8a63df2e837386.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.551ex; height:2.843ex;" alt="{\displaystyle (X(1)\mid X(0)=i)}" loading="lazy"></span> are identically distributed, so their variances are equal. Writing as before <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=X(t)}">

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<annotation encoding="application/x-tex">{\displaystyle Z=X(t-1)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {Var} (Y)&=\operatorname {E} [\operatorname {Var} (Y\mid Z)]+\operatorname {Var} (\operatorname {E} [Y\mid Z])\\&=E\left[\left({\frac {2Z}{N}}\right)\left(1-{\frac {Z}{N}}\right)\right]+\operatorname {Var} (Z)\\&=\left({\frac {2\operatorname {E} [Z]}{N}}\right)\left(1-{\frac {\operatorname {E} [Z]}{N}}\right)+\left(1-{\frac {2}{N^{2}}}\right)\operatorname {Var} (Z).\end{aligned}}}</annotation>

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</math></span><img src="./31bf4288b48d0f9e079a00d9d0ab19a13d63c60e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:57.436ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {Var} (Y)&=\operatorname {E} [\operatorname {Var} (Y\mid Z)]+\operatorname {Var} (\operatorname {E} [Y\mid Z])\\&=E\left[\left({\frac {2Z}{N}}\right)\left(1-{\frac {Z}{N}}\right)\right]+\operatorname {Var} (Z)\\&=\left({\frac {2\operatorname {E} [Z]}{N}}\right)\left(1-{\frac {\operatorname {E} [Z]}{N}}\right)+\left(1-{\frac {2}{N^{2}}}\right)\operatorname {Var} (Z).\end{aligned}}}" loading="lazy"></span></dd></dl>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle X(0)=i}</annotation>

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</math></span><img src="./98392b5bd4a308e3788f972df7109310c519c07e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.853ex; height:2.843ex;" alt="{\displaystyle X(0)=i}" loading="lazy"></span>, we obtain

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<annotation encoding="application/x-tex">{\displaystyle V_{t}=V_{1}+\left(1-{\frac {2}{N^{2}}}\right)V_{t-1}.}</annotation>

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</math></span><img src="./fb93bf2c007086430264441c43a12b335e15e90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.283ex; height:6.176ex;" alt="{\displaystyle V_{t}=V_{1}+\left(1-{\frac {2}{N^{2}}}\right)V_{t-1}.}" loading="lazy"></span></dd></dl>

<p>Rewriting this equation as

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<annotation encoding="application/x-tex">{\displaystyle V_{t}-{\frac {V_{1}}{\frac {2}{N^{2}}}}=\left(1-{\frac {2}{N^{2}}}\right)\left(V_{t-1}-{\frac {V_{1}}{\frac {2}{N^{2}}}}\right)=\left(1-{\frac {2}{N^{2}}}\right)^{t-1}\left(V_{1}-{\frac {V_{1}}{\frac {2}{N^{2}}}}\right)}</annotation>

</semantics>

</math></span><img src="./8c57c265a999620a7ed980f68165200b6a111518.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:70.315ex; height:8.843ex;" alt="{\displaystyle V_{t}-{\frac {V_{1}}{\frac {2}{N^{2}}}}=\left(1-{\frac {2}{N^{2}}}\right)\left(V_{t-1}-{\frac {V_{1}}{\frac {2}{N^{2}}}}\right)=\left(1-{\frac {2}{N^{2}}}\right)^{t-1}\left(V_{1}-{\frac {V_{1}}{\frac {2}{N^{2}}}}\right)}" loading="lazy"></span></dd></dl>

<p>yields

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<annotation encoding="application/x-tex">{\displaystyle V_{t}=V_{1}{\frac {1-\left(1-{\frac {2}{N^{2}}}\right)^{t}}{\frac {2}{N^{2}}}}}</annotation>

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</math></span><img src="./481015f601ff70834225d79081809dbfa9efebd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:23.301ex; height:9.843ex;" alt="{\displaystyle V_{t}=V_{1}{\frac {1-\left(1-{\frac {2}{N^{2}}}\right)^{t}}{\frac {2}{N^{2}}}}}" loading="lazy"></span></dd></dl>

<p>as desired.

</p>

<p>The probability that A reaches fixation is called <i>fixation probability</i>. For the simple Moran process this probability is <span class="texhtml"><i>x<sub>i</sub></i> = <span class="sfrac">⁠<span class="tion"><span class="num"><i>i</i></span><span class="sr-only">/</span><span class="den"><i>N</i></span></span>⁠</span>.</span>

</p><p>Since all individuals have the same fitness, they also have the same chance of becoming the ancestor of the whole population; this probability is <span class="texhtml"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>N</i></span></span>⁠</span></span> and thus the sum of all <i>i</i> probabilities (for all A individuals) is just <span class="texhtml"><span class="sfrac">⁠<span class="tion"><span class="num"><i>i</i></span><span class="sr-only">/</span><span class="den"><i>N</i></span></span>⁠</span>.</span> The mean time to absorption starting in state <i>i</i> is given by

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<annotation encoding="application/x-tex">{\displaystyle k_{i}=N\left[\sum _{j=1}^{i}{\frac {N-i}{N-j}}+\sum _{j=i+1}^{N-1}{\frac {i}{j}}\right]}</annotation>

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</math></span><img src="./1357d063b01000efd7f85646e6721385b9245efd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.356ex; height:7.676ex;" alt="{\displaystyle k_{i}=N\left[\sum _{j=1}^{i}{\frac {N-i}{N-j}}+\sum _{j=i+1}^{N-1}{\frac {i}{j}}\right]}" loading="lazy"></span></dd></dl>

<div class="hidden-begin mw-collapsible mw-collapsed" style=""><div class="hidden-title skin-nightmode-reset-color" style="text-align:center; background-color:#f2dfce; color:black;">For a mathematical derivation of the equation above, click on "show" to reveal</div><div class="hidden-content mw-collapsible-content" style="border:1px #C4C3D0 solid;">

<p>The mean time spent in state <i>j</i> when starting in state <i>i</i> which is given by

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<annotation encoding="application/x-tex">{\displaystyle k_{i}^{j}=\delta _{ij}+P_{i,i-1}k_{i-1}^{j}+P_{i,i}k_{i}^{j}+P_{i,i+1}k_{i+1}^{j}}</annotation>

</semantics>

</math></span><img src="./3157f76d32860e428d9f97cdffb34ea130189165.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:40.745ex; height:3.676ex;" alt="{\displaystyle k_{i}^{j}=\delta _{ij}+P_{i,i-1}k_{i-1}^{j}+P_{i,i}k_{i}^{j}+P_{i,i+1}k_{i+1}^{j}}" loading="lazy"></span></dd></dl>

<p>Here <span class="texhtml mvar" style="font-style:italic;">δ<sub>ij</sub></span> denotes the <a href="Kroenecker_delta" class="mw-redirect" title="Kroenecker delta">Kroenecker delta</a>. This <a href="Recurrence_relation" title="Recurrence relation">recursive equation</a> can be solved using a new variable <span class="texhtml mvar" style="font-style:italic;">q<sub>i</sub></span> so that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i,i-1}=P_{i,i+1}=q_{i}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle P_{i,i-1}=P_{i,i+1}=q_{i}}</annotation>

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</math></span><img src="./f72b7820d579a9cfdd6d489f649e0df9f111f67a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.868ex; height:2.843ex;" alt="{\displaystyle P_{i,i-1}=P_{i,i+1}=q_{i}}" loading="lazy"></span> and thus <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i,i}=1-2q_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{i,i}=1-2q_{i}}</annotation>

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</math></span><img src="./3de81883d5b6a01b68dbb60fc68641057e999114.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.417ex; height:2.843ex;" alt="{\displaystyle P_{i,i}=1-2q_{i}}" loading="lazy"></span> and rewritten

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<annotation encoding="application/x-tex">{\displaystyle k_{i+1}^{j}=2k_{i}^{j}-k_{i-1}^{j}-{\frac {\delta _{ij}}{q_{i}}}}</annotation>

</semantics>

</math></span><img src="./2468bbda88c4f4bbf3f9c1931a814736fe18a950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.631ex; height:6.176ex;" alt="{\displaystyle k_{i+1}^{j}=2k_{i}^{j}-k_{i-1}^{j}-{\frac {\delta _{ij}}{q_{i}}}}" loading="lazy"></span></dd></dl>

<p>The variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}^{j}=k_{i}^{j}-k_{i-1}^{j}}">

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<annotation encoding="application/x-tex">{\displaystyle y_{i}^{j}=k_{i}^{j}-k_{i-1}^{j}}</annotation>

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</math></span><img src="./fc153b1b91ae08a8f0496f3553af3a1927f2a24c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.241ex; height:3.676ex;" alt="{\displaystyle y_{i}^{j}=k_{i}^{j}-k_{i-1}^{j}}" loading="lazy"></span> is used and the equation becomes

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y_{i+1}^{j}&=y_{i}^{j}-{\frac {\delta _{ij}}{q_{i}}}\\\\\sum _{i=1}^{m}y_{i}^{j}&=(k_{1}^{j}-k_{0}^{j})+(k_{2}^{j}-k_{1}^{j})+\cdots +(k_{m-1}^{j}-k_{m-2}^{j})+(k_{m}^{j}-k_{m-1}^{j})\\&=k_{m}^{j}-k_{0}^{j}\\\sum _{i=1}^{m}y_{i}^{j}&=k_{m}^{j}\\\\y_{1}^{j}&=(k_{1}^{j}-k_{0}^{j})=k_{1}^{j}\\y_{2}^{j}&=y_{1}^{j}-{\frac {\delta _{1j}}{q_{1}}}=k_{1}^{j}-{\frac {\delta _{1j}}{q_{1}}}\\y_{3}^{j}&=k_{1}^{j}-{\frac {\delta _{1j}}{q_{1}}}-{\frac {\delta _{2j}}{q_{2}}}\\&\vdots \\y_{i}^{j}&=k_{1}^{j}-\sum _{r=1}^{i-1}{\frac {\delta _{rj}}{q_{r}}}={\begin{cases}k_{1}^{j}&j\geq i\\k_{1}^{j}-{\frac {1}{q_{j}}}&j\leq i\end{cases}}\\\\k_{i}^{j}&=\sum _{m=1}^{i}y_{m}^{j}={\begin{cases}i\cdot k_{1}^{j}&j\geq i\\i\cdot k_{1}^{j}-{\frac {i-j}{q_{j}}}&j\leq i\end{cases}}\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./ccda6fdbee86de55e50e0b6b2a8740272ce77855.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -34.005ex; width:69.254ex; height:69.176ex;" alt="{\displaystyle {\begin{aligned}y_{i+1}^{j}&=y_{i}^{j}-{\frac {\delta _{ij}}{q_{i}}}\\\\\sum _{i=1}^{m}y_{i}^{j}&=(k_{1}^{j}-k_{0}^{j})+(k_{2}^{j}-k_{1}^{j})+\cdots +(k_{m-1}^{j}-k_{m-2}^{j})+(k_{m}^{j}-k_{m-1}^{j})\\&=k_{m}^{j}-k_{0}^{j}\\\sum _{i=1}^{m}y_{i}^{j}&=k_{m}^{j}\\\\y_{1}^{j}&=(k_{1}^{j}-k_{0}^{j})=k_{1}^{j}\\y_{2}^{j}&=y_{1}^{j}-{\frac {\delta _{1j}}{q_{1}}}=k_{1}^{j}-{\frac {\delta _{1j}}{q_{1}}}\\y_{3}^{j}&=k_{1}^{j}-{\frac {\delta _{1j}}{q_{1}}}-{\frac {\delta _{2j}}{q_{2}}}\\&\vdots \\y_{i}^{j}&=k_{1}^{j}-\sum _{r=1}^{i-1}{\frac {\delta _{rj}}{q_{r}}}={\begin{cases}k_{1}^{j}&j\geq i\\k_{1}^{j}-{\frac {1}{q_{j}}}&j\leq i\end{cases}}\\\\k_{i}^{j}&=\sum _{m=1}^{i}y_{m}^{j}={\begin{cases}i\cdot k_{1}^{j}&j\geq i\\i\cdot k_{1}^{j}-{\frac {i-j}{q_{j}}}&j\leq i\end{cases}}\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>Knowing that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{N}^{j}=0}">

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<annotation encoding="application/x-tex">{\displaystyle k_{N}^{j}=0}</annotation>

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</math></span><img src="./b51532ee41a27f5bc4dcaccc62323420dcfe1d3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.164ex; height:3.509ex;" alt="{\displaystyle k_{N}^{j}=0}" loading="lazy"></span> and

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<annotation encoding="application/x-tex">{\displaystyle q_{j}=P_{j,j+1}={\frac {j}{N}}{\frac {N-j}{N}}}</annotation>

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</math></span><img src="./e00fd959c476e54ce2f10dfb75d5fdc676dd9b0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.379ex; height:5.343ex;" alt="{\displaystyle q_{j}=P_{j,j+1}={\frac {j}{N}}{\frac {N-j}{N}}}" loading="lazy"></span></dd></dl>

<p>we can calculate <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}^{j}}">

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<annotation encoding="application/x-tex">{\displaystyle k_{1}^{j}}</annotation>

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</math></span><img src="./9122adf530c67daae524746389c18c744b344fea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.265ex; height:3.509ex;" alt="{\displaystyle k_{1}^{j}}" loading="lazy"></span>:

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</msubsup>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</msubsup>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}k_{N}^{j}=\sum _{i=1}^{m}y_{i}^{j}=N\cdot k_{1}^{j}&-{\frac {N-j}{q_{j}}}=0\\k_{1}^{j}&={\frac {N}{j}}\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./69704c3b8982a2e37587c4b6726c65a8e70c7314.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:35.471ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}k_{N}^{j}=\sum _{i=1}^{m}y_{i}^{j}=N\cdot k_{1}^{j}&-{\frac {N-j}{q_{j}}}=0\\k_{1}^{j}&={\frac {N}{j}}\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>Therefore

</p>

</mrow>

</mrow>

</msubsup>

<mrow>

<mtr>

<mtd>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k_{i}^{j}={\begin{cases}{\frac {i}{j}}\cdot k_{j}^{j}&j\geq i\\{\frac {N-i}{N-j}}\cdot k_{j}^{j}&j\leq i\end{cases}}}</annotation>

</semantics>

</math></span><img src="./57bf70c15c3e07f9346f3de875b043ba93ef1b54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:23.077ex; height:8.176ex;" alt="{\displaystyle k_{i}^{j}={\begin{cases}{\frac {i}{j}}\cdot k_{j}^{j}&j\geq i\\{\frac {N-i}{N-j}}\cdot k_{j}^{j}&j\leq i\end{cases}}}" loading="lazy"></span></dd></dl>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k_{j}^{j}=N}</annotation>

</semantics>

</math></span><img src="./970280a6f95b2b33e230b98bf7e4a3f6c73bc0b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.283ex; height:3.843ex;" alt="{\displaystyle k_{j}^{j}=N}" loading="lazy"></span>. Now <span class="texhtml mvar" style="font-style:italic;">k<sub>i</sub></span>, the total time until fixation starting from state <i>i</i>, can be calculated

</p>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</msubsup>

</mtd>

<mtd>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</msubsup>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}k_{i}=\sum _{j=1}^{N-1}k_{i}^{j}&=\sum _{j=1}^{i}k_{i}^{j}+\sum _{j=i+1}^{N-1}k_{i}^{j}\\&=\sum _{j=1}^{i}N{\frac {N-i}{N-j}}+\sum _{j=i+1}^{N-1}N{\frac {i}{j}}\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./3ca311ef8d0cce4a76bbe2bb25b52c14c007c340.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:39.24ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}k_{i}=\sum _{j=1}^{N-1}k_{i}^{j}&=\sum _{j=1}^{i}k_{i}^{j}+\sum _{j=i+1}^{N-1}k_{i}^{j}\\&=\sum _{j=1}^{i}N{\frac {N-i}{N-j}}+\sum _{j=i+1}^{N-1}N{\frac {i}{j}}\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>For large <i>N</i> the approximation

</p>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{N\to \infty }k_{i}\approx -N^{2}\left[(1-x_{i})\ln(1-x_{i})+x_{i}\ln(x_{i})\right]}</annotation>

</semantics>

</math></span><img src="./43d810eb747e78b1286393e2cc514839179c9eb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:46.352ex; height:4.343ex;" alt="{\displaystyle \lim _{N\to \infty }k_{i}\approx -N^{2}\left[(1-x_{i})\ln(1-x_{i})+x_{i}\ln(x_{i})\right]}" loading="lazy"></span></dd></dl>

<p>holds.

</p>

<div class="mw-heading mw-heading2"><h2 id="Selection">Selection</h2></div>

<p>If one allele has a <a href="Fitness_(biology)" title="Fitness (biology)">fitness advantage</a> over the other allele, it will be more likely to be chosen for reproduction. This can be incorporated into the model if individuals with <a href="Allele" title="Allele">allele</a> A have fitness <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}>0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{i}>0}</annotation>

</semantics>

</math></span><img src="./02b5194db579f9562e86c6456c0dbf15a6f7feee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.2ex; height:2.509ex;" alt="{\displaystyle f_{i}>0}" loading="lazy"></span> and individuals with <a href="Allele" title="Allele">allele</a> B have fitness <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{i}>0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{i}>0}</annotation>

</semantics>

</math></span><img src="./f827d07cbef705feab3481e7aa62717111228ab1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.17ex; height:2.509ex;" alt="{\displaystyle g_{i}>0}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> is the number of individuals of type A; thus describing a general birth-death process. The transition matrix of the stochastic process is <a href="Tridiagonal_matrix" title="Tridiagonal matrix">tri-diagonal</a> in shape. Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}:=f_{i}/g_{i}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{i}:=f_{i}/g_{i}}</annotation>

</semantics>

</math></span><img src="./479b5bebdec0a9a23bd6bb8c127d659acbd2ba2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.604ex; height:2.843ex;" alt="{\displaystyle r_{i}:=f_{i}/g_{i}}" loading="lazy"></span>, then the transition probabilities are

</p>

<mtr>

<mtd>

<msub>

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</mtd>

<mtd>

<mfrac>

<mrow>

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<mrow>

<msub>

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</msub>

<msub>

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</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

<mrow>

<msub>

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</msub>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

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</mfrac>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

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<mtd>

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</mtr>

<mtr>

<mtd>

<msub>

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</msub>

</mtd>

<mtd>

<mfrac>

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</msub>

</mrow>

<mrow>

<msub>

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</msub>

<msub>

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</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

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</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mrow>

<msub>

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</msub>

<mfrac>

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</mrow>

<mfrac>

<mrow>

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</mfrac>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P_{i,i-1}&={\frac {g_{i}\cdot (N-i)}{f_{i}\cdot i+g_{i}\cdot (N-i)}}\cdot {\frac {i}{N}}={\frac {\frac {N-i}{N}}{r_{i}\cdot {\frac {i}{N}}+{\frac {N-i}{N}}}}\cdot {\frac {i}{N}}\\P_{i,i}&=1-P_{i,i-1}-P_{i,i+1}\\P_{i,i+1}&={\frac {f_{i}\cdot i}{f_{i}\cdot i+g_{i}\cdot (N-i)}}\cdot {\frac {N-i}{N}}={\frac {r_{i}\cdot {\frac {i}{N}}}{r_{i}\cdot {\frac {i}{N}}+{\frac {N-i}{N}}}}\cdot {\frac {N-i}{N}}\\\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./7e891f8be8e8dd01e777d50981f47917fa64c472.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.338ex; width:61.65ex; height:19.843ex;" alt="{\displaystyle {\begin{aligned}P_{i,i-1}&={\frac {g_{i}\cdot (N-i)}{f_{i}\cdot i+g_{i}\cdot (N-i)}}\cdot {\frac {i}{N}}={\frac {\frac {N-i}{N}}{r_{i}\cdot {\frac {i}{N}}+{\frac {N-i}{N}}}}\cdot {\frac {i}{N}}\\P_{i,i}&=1-P_{i,i-1}-P_{i,i+1}\\P_{i,i+1}&={\frac {f_{i}\cdot i}{f_{i}\cdot i+g_{i}\cdot (N-i)}}\cdot {\frac {N-i}{N}}={\frac {r_{i}\cdot {\frac {i}{N}}}{r_{i}\cdot {\frac {i}{N}}+{\frac {N-i}{N}}}}\cdot {\frac {N-i}{N}}\\\end{aligned}}}" loading="lazy"></span></dd></dl>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{i,j}}</annotation>

</semantics>

</p>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {r_{i}\cdot {\frac {i}{N}}}{r_{i}\cdot {\frac {i}{N}}+{\frac {N-i}{i}}}},}</annotation>

</semantics>

</math></span><img src="./440f650dc5ee244012fcfc948623d6b972324516.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:14.287ex; height:8.343ex;" alt="{\displaystyle {\frac {r_{i}\cdot {\frac {i}{N}}}{r_{i}\cdot {\frac {i}{N}}+{\frac {N-i}{i}}}},}" loading="lazy"></span></dd></dl>

<p>and an individual with allele B is chosen with probability

</p>

<mfrac>

<mrow>

</mrow>

</mfrac>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\frac {N-i}{N}}{r_{i}\cdot {\frac {i}{N}}+{\frac {N-i}{i}}}},}</annotation>

</semantics>

</math></span><img src="./7fe89e199e2f68e34a3241b03cafa7e63ff6a98b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:14.287ex; height:8.343ex;" alt="{\displaystyle {\frac {\frac {N-i}{N}}{r_{i}\cdot {\frac {i}{N}}+{\frac {N-i}{i}}}},}" loading="lazy"></span></dd></dl>

<p>when the number of individuals with allele A is exactly i.

</p><p>Also in this case, fixation probabilities when starting in state <i>i</i> is defined by the recurrence

</p>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

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</msub>

<msub>

</mrow>

</msub>

<msub>

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</msub>

<msub>

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</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

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</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}={\begin{cases}0&i=0\\\beta _{i}x_{i-1}+(1-\alpha _{i}-\beta _{i})x_{i}+\alpha _{i}x_{i+1}&1\leq i\leq N-1\\1&i=N\end{cases}}}</annotation>

</semantics>

</math></span><img src="./419fadae6e5a47b930ce42366b5f87b32928992f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:58.324ex; height:8.509ex;" alt="{\displaystyle x_{i}={\begin{cases}0&i=0\\\beta _{i}x_{i-1}+(1-\alpha _{i}-\beta _{i})x_{i}+\alpha _{i}x_{i+1}&1\leq i\leq N-1\\1&i=N\end{cases}}}" loading="lazy"></span></dd></dl>

<p>And the closed form is given by

</p>

<msub>

</mrow>

</msub>

<mfrac>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}={\frac {\displaystyle 1+\sum _{j=1}^{i-1}\prod _{k=1}^{j}\gamma _{k}}{\displaystyle 1+\sum _{j=1}^{N-1}\prod _{k=1}^{j}\gamma _{k}}}\qquad {\text{(1)}}}</annotation>

</semantics>

</math></span><img src="./06801c2341dce81fdf878eca4c4dd2eaed0e8ede.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:27.28ex; height:16.176ex;" alt="{\displaystyle x_{i}={\frac {\displaystyle 1+\sum _{j=1}^{i-1}\prod _{k=1}^{j}\gamma _{k}}{\displaystyle 1+\sum _{j=1}^{N-1}\prod _{k=1}^{j}\gamma _{k}}}\qquad {\text{(1)}}}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{i}=P_{i,i-1}/P_{i,i+1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{i}=P_{i,i-1}/P_{i,i+1}}</annotation>

</semantics>

</math></span><img src="./777191c9d468ce5c9f4dad61d59158e6bca2f671.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.099ex; height:3.009ex;" alt="{\displaystyle \gamma _{i}=P_{i,i-1}/P_{i,i+1}}" loading="lazy"></span> per definition and will just be <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{i}/f_{i}}">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{i}/f_{i}}</annotation>

</semantics>

</math></span><img src="./c2cda28ead8f6340223de17165d57b3a3bb10f67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.01ex; height:2.843ex;" alt="{\displaystyle g_{i}/f_{i}}" loading="lazy"></span> for the general case.

</p>

<p>Also in this case, fixation probabilities can be computed, but the transition probabilities are not symmetric. The notation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i,i+1}=\alpha _{i},P_{i,i-1}=\beta _{i},P_{i,i}=1-\alpha _{i}-\beta _{i}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{i,i+1}=\alpha _{i},P_{i,i-1}=\beta _{i},P_{i,i}=1-\alpha _{i}-\beta _{i}}</annotation>

</semantics>

</math></span><img src="./7524ec9844513c60316d4aac7e5b22e28d8693bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.163ex; height:2.843ex;" alt="{\displaystyle P_{i,i+1}=\alpha _{i},P_{i,i-1}=\beta _{i},P_{i,i}=1-\alpha _{i}-\beta _{i}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{i}=\beta _{i}/\alpha _{i}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{i}=\beta _{i}/\alpha _{i}}</annotation>

</semantics>

</math></span><img src="./7cd1c5eb4fb5fb4873b34c2c9218c5e731f4beaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.668ex; height:2.843ex;" alt="{\displaystyle \gamma _{i}=\beta _{i}/\alpha _{i}}" loading="lazy"></span> is used. The fixation probability can be defined recursively and a new variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}=x_{i}-x_{i-1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle y_{i}=x_{i}-x_{i-1}}</annotation>

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</math></span><img src="./16af3c5f1c40e550b1d67a04901a12e1cd3467f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.237ex; height:2.343ex;" alt="{\displaystyle y_{i}=x_{i}-x_{i-1}}" loading="lazy"></span> is introduced.

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{i}&=\beta _{i}x_{i-1}+(1-\alpha _{i}-\beta _{i})x_{i}+\alpha _{i}x_{i+1}\\\beta _{i}(x_{i}-x_{i-1})&=\alpha _{i}(x_{i+1}-x_{i})\\\gamma _{i}\cdot y_{i}&=y_{i+1}\end{aligned}}}</annotation>

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</math></span><img src="./e63a4920e33c6e80b8f2faad098e69456f4ffc81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:50.702ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}x_{i}&=\beta _{i}x_{i-1}+(1-\alpha _{i}-\beta _{i})x_{i}+\alpha _{i}x_{i+1}\\\beta _{i}(x_{i}-x_{i-1})&=\alpha _{i}(x_{i+1}-x_{i})\\\gamma _{i}\cdot y_{i}&=y_{i+1}\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>Now two properties from the definition of the variable <span class="texhtml mvar" style="font-style:italic;">y<sub>i</sub></span> can be used to find a closed form solution for the fixation probabilities:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{i=1}^{m}y_{i}&=x_{m}&&1\\y_{k}&=x_{1}\cdot \prod _{l=1}^{k-1}\gamma _{l}&&2\\\Rightarrow \sum _{m=1}^{i}y_{m}&=x_{1}+x_{1}\sum _{j=1}^{i-1}\prod _{k=1}^{j}\gamma _{k}=x_{i}&&3\end{aligned}}}</annotation>

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</math></span><img src="./79ca3479a75f3dbfae1fd8bb923768d6b44f5b46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.671ex; width:41.986ex; height:22.509ex;" alt="{\displaystyle {\begin{aligned}\sum _{i=1}^{m}y_{i}&=x_{m}&&1\\y_{k}&=x_{1}\cdot \prod _{l=1}^{k-1}\gamma _{l}&&2\\\Rightarrow \sum _{m=1}^{i}y_{m}&=x_{1}+x_{1}\sum _{j=1}^{i-1}\prod _{k=1}^{j}\gamma _{k}=x_{i}&&3\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>Combining (3) and <span class="texhtml"><i>x<sub>N</sub></i> = 1</span>:

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<annotation encoding="application/x-tex">{\displaystyle x_{1}\left(1+\sum _{j=1}^{N-1}\prod _{k=1}^{j}\gamma _{k}\right)=x_{N}=1.}</annotation>

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</math></span><img src="./118e512b2e76e5fd4fe11b276944001bee184e94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:31.079ex; height:7.676ex;" alt="{\displaystyle x_{1}\left(1+\sum _{j=1}^{N-1}\prod _{k=1}^{j}\gamma _{k}\right)=x_{N}=1.}" loading="lazy"></span></dd></dl>

<p>which implies:

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<annotation encoding="application/x-tex">{\displaystyle x_{1}={\frac {1}{1+\sum _{j=1}^{N-1}\prod _{k=1}^{j}\gamma _{k}}}}</annotation>

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</math></span><img src="./2ad6fa419d2d6e35c75573f1c8a4f0e1d275b3a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:25.017ex; height:6.843ex;" alt="{\displaystyle x_{1}={\frac {1}{1+\sum _{j=1}^{N-1}\prod _{k=1}^{j}\gamma _{k}}}}" loading="lazy"></span></dd></dl>

<p>This in turn gives us:

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</math></span><img src="./5758fd0e15f43df8cb6b8168a366ec7196e4d7a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:19.663ex; height:16.176ex;" alt="{\displaystyle x_{i}={\frac {\displaystyle 1+\sum _{j=1}^{i-1}\prod _{k=1}^{j}\gamma _{k}}{\displaystyle 1+\sum _{j=1}^{N-1}\prod _{k=1}^{j}\gamma _{k}}}}" loading="lazy"></span></dd></dl>

<p>This general case where the fitness of A and B depends on the abundance of each type is studied in <a href="Evolutionary_game_theory" title="Evolutionary game theory">evolutionary game theory</a>.

</p><p>Less complex results are obtained if a constant fitness ratio <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=1/\gamma _{i}}">

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<annotation encoding="application/x-tex">{\displaystyle r=1/\gamma _{i}}</annotation>

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</math></span><img src="./1e6b555d5473c418d6a243bb933ede914416fb7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.476ex; height:2.843ex;" alt="{\displaystyle r=1/\gamma _{i}}" loading="lazy"></span>, for all i, is assumed. Individuals of type A reproduce with a constant rate <i>r</i> and individuals with allele B reproduce with rate 1. Thus if A has a fitness advantage over B, <i>r</i> will be larger than one, otherwise it will be smaller than one. Thus the transition matrix of the stochastic process is tri-diagonal in shape and the transition probabilities are

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P_{0,0}&=1\\P_{i,i-1}&={\frac {N-i}{r\cdot i+N-i}}\cdot {\frac {i}{N}}={\frac {\frac {N-i}{N}}{r\cdot {\frac {i}{N}}+{\frac {N-i}{N}}}}\cdot {\frac {i}{N}}\\P_{i,i}&=1-P_{i,i-1}-P_{i,i+1}\\P_{i,i+1}&={\frac {r\cdot i}{r\cdot i+N-i}}\cdot {\frac {N-i}{N}}={\frac {r\cdot {\frac {i}{N}}}{r\cdot {\frac {i}{N}}+{\frac {N-i}{N}}}}\cdot {\frac {N-i}{N}}\\P_{N,N}&=1.\end{aligned}}}</annotation>

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</math></span><img src="./92457809cb27b8f6461781019a3a1575049f81bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.671ex; width:54.563ex; height:26.509ex;" alt="{\displaystyle {\begin{aligned}P_{0,0}&=1\\P_{i,i-1}&={\frac {N-i}{r\cdot i+N-i}}\cdot {\frac {i}{N}}={\frac {\frac {N-i}{N}}{r\cdot {\frac {i}{N}}+{\frac {N-i}{N}}}}\cdot {\frac {i}{N}}\\P_{i,i}&=1-P_{i,i-1}-P_{i,i+1}\\P_{i,i+1}&={\frac {r\cdot i}{r\cdot i+N-i}}\cdot {\frac {N-i}{N}}={\frac {r\cdot {\frac {i}{N}}}{r\cdot {\frac {i}{N}}+{\frac {N-i}{N}}}}\cdot {\frac {N-i}{N}}\\P_{N,N}&=1.\end{aligned}}}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \gamma _{i}=1/r}</annotation>

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</math></span><img src="./a60e38a25d49e6390734343b9b90e50515deb6c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.476ex; height:2.843ex;" alt="{\displaystyle \gamma _{i}=1/r}" loading="lazy"></span> is a constant factor for each composition of the population and thus the fixation probability from equation (1) simplifies to

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<annotation encoding="application/x-tex">{\displaystyle x_{i}={\frac {1-r^{-i}}{1-r^{-N}}}\quad \Rightarrow \quad x_{1}=\rho ={\frac {1-r^{-1}}{1-r^{-N}}}\qquad {\text{(2)}}}</annotation>

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</math></span><img src="./608185c87a518607340320b6e8bcc0b2571420a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:48.602ex; height:6.176ex;" alt="{\displaystyle x_{i}={\frac {1-r^{-i}}{1-r^{-N}}}\quad \Rightarrow \quad x_{1}=\rho ={\frac {1-r^{-1}}{1-r^{-N}}}\qquad {\text{(2)}}}" loading="lazy"></span></dd></dl>

<p>where the fixation probability of a single mutant <i>A</i> in a population of otherwise all <i>B</i> is often of interest and is denoted by <span class="texhtml mvar" style="font-style:italic;">ρ</span>.

</p><p>Also in the case of selection, the expected value and the variance of the number of <i>A</i> individuals may be computed

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} [X(t)\mid X(t-1)=i]&=ps{\dfrac {1-p}{ps+1}}+i\\\operatorname {Var} (X(t+1)\mid X(t)=i)&=p(1-p){\dfrac {(s+1)+(ps+1)^{2}}{(ps+1)^{2}}}\end{aligned}}}</annotation>

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</math></span><img src="./b6ea29b762c1eb9043657d71dc17602f13fdd28d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.457ex; margin-bottom: -0.215ex; width:56.432ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {E} [X(t)\mid X(t-1)=i]&=ps{\dfrac {1-p}{ps+1}}+i\\\operatorname {Var} (X(t+1)\mid X(t)=i)&=p(1-p){\dfrac {(s+1)+(ps+1)^{2}}{(ps+1)^{2}}}\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>where <span class="texhtml"><i>p</i> = <span class="sfrac">⁠<span class="tion"><span class="num"><i>i</i></span><span class="sr-only">/</span><span class="den"><i>N</i></span></span>⁠</span>,</span> and <span class="texhtml"><i>r</i> = 1 + <i>s</i></span>.

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<p>For the expected value the calculation runs as follows

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} [\Delta (1)\mid X(0)=i]&=(i-1-i)\cdot P_{i,i-1}+(i-i)\cdot P_{i,i}+(i+1-i)\cdot P_{i,i+1}\\&=-{\frac {N-i}{ri+N-i}}{\frac {i}{N}}+{\frac {ri}{ri+N-i}}{\frac {N-i}{N}}\\&=-{\frac {(N-i)i}{(ri+N-i)N}}+{\frac {i(N-i)}{(ri+N-i)N}}+{\frac {si(N-i)}{(ri+N-i)N}}\\&=ps{\dfrac {1-p}{ps+1}}\\\operatorname {E} [X(t)\mid X(t-1)=i]&=ps{\dfrac {1-p}{ps+1}}+i\end{aligned}}}</annotation>

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</math></span><img src="./91e18f52a973196307fdc232d30f341f6d49b461.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.005ex; width:78.635ex; height:27.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {E} [\Delta (1)\mid X(0)=i]&=(i-1-i)\cdot P_{i,i-1}+(i-i)\cdot P_{i,i}+(i+1-i)\cdot P_{i,i+1}\\&=-{\frac {N-i}{ri+N-i}}{\frac {i}{N}}+{\frac {ri}{ri+N-i}}{\frac {N-i}{N}}\\&=-{\frac {(N-i)i}{(ri+N-i)N}}+{\frac {i(N-i)}{(ri+N-i)N}}+{\frac {si(N-i)}{(ri+N-i)N}}\\&=ps{\dfrac {1-p}{ps+1}}\\\operatorname {E} [X(t)\mid X(t-1)=i]&=ps{\dfrac {1-p}{ps+1}}+i\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>For the variance the calculation runs as follows, using the variance of a single step

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {Var} (X(t+1)\mid X(t)=i)&=\operatorname {Var} (X(t))+\operatorname {Var} (\Delta (t+1)\mid X(t)=i)\\&=0+E\left[\Delta (t+1)^{2}\mid X(t)=i\right]-\operatorname {E} [\Delta (t+1)\mid X(t)=i]^{2}\\&=(i-1-i)^{2}\cdot P_{i,i-1}+(i-i)^{2}\cdot P_{i,i}+(i+1-i)^{2}\cdot P_{i,i+1}-\operatorname {E} [\Delta (t+1)\mid X(t)=i]^{2}\\&=P_{i,i-1}+P_{i,i+1}-\operatorname {E} [\Delta (t+1)\mid X(t)=i]^{2}\\&={\frac {(N-i)i}{(ri+N-i)N}}+{\frac {(N-i)i(1+s)}{(ri+N-i)N}}-\operatorname {E} [\Delta (t+1)\mid X(t)=i]^{2}\\&=i(N-i){\frac {2+s}{(ri+N-i)N}}-\operatorname {E} [\Delta (t+1)\mid X(t)=i]^{2}\\&=i(N-i){\frac {2+s}{(ri+N-i)N}}-\left(ps{\dfrac {1-p}{ps+1}}\right)^{2}\\&=p(1-p){\frac {2+s(ps+1)}{(ps+1)^{2}}}-p(1-p){\frac {ps^{2}(1-p)}{(ps+1)^{2}}}\\&=p(1-p){\dfrac {2+2ps+s+p^{2}s^{2}}{(ps+1)^{2}}}\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./954758138ecde3d084d234bf57d6a96f635dbf66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -22.671ex; width:109.203ex; height:46.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {Var} (X(t+1)\mid X(t)=i)&=\operatorname {Var} (X(t))+\operatorname {Var} (\Delta (t+1)\mid X(t)=i)\\&=0+E\left[\Delta (t+1)^{2}\mid X(t)=i\right]-\operatorname {E} [\Delta (t+1)\mid X(t)=i]^{2}\\&=(i-1-i)^{2}\cdot P_{i,i-1}+(i-i)^{2}\cdot P_{i,i}+(i+1-i)^{2}\cdot P_{i,i+1}-\operatorname {E} [\Delta (t+1)\mid X(t)=i]^{2}\\&=P_{i,i-1}+P_{i,i+1}-\operatorname {E} [\Delta (t+1)\mid X(t)=i]^{2}\\&={\frac {(N-i)i}{(ri+N-i)N}}+{\frac {(N-i)i(1+s)}{(ri+N-i)N}}-\operatorname {E} [\Delta (t+1)\mid X(t)=i]^{2}\\&=i(N-i){\frac {2+s}{(ri+N-i)N}}-\operatorname {E} [\Delta (t+1)\mid X(t)=i]^{2}\\&=i(N-i){\frac {2+s}{(ri+N-i)N}}-\left(ps{\dfrac {1-p}{ps+1}}\right)^{2}\\&=p(1-p){\frac {2+s(ps+1)}{(ps+1)^{2}}}-p(1-p){\frac {ps^{2}(1-p)}{(ps+1)^{2}}}\\&=p(1-p){\dfrac {2+2ps+s+p^{2}s^{2}}{(ps+1)^{2}}}\end{aligned}}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Rate_of_evolution">Rate of evolution</h2></div>

<p>In a population of all <i>B</i> individuals, a single mutant <i>A</i> will take over the whole population with the probability

</p>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {1-r^{-1}}{1-r^{-N}}}.\qquad {\text{(2)}}}</annotation>

</semantics>

</math></span><img src="./b06d417bf0456530c2076fdde1dcb043070e2d5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.809ex; height:6.176ex;" alt="{\displaystyle \rho ={\frac {1-r^{-1}}{1-r^{-N}}}.\qquad {\text{(2)}}}" loading="lazy"></span></dd></dl>

<p>If the <a href="Mutation_rate" title="Mutation rate">mutation rate</a> (to go from the <i>B</i> to the <i>A</i> allele) in the population is <i>u</i> then the rate with which one member of the population will mutate to <i>A</i> is given by <span class="texhtml"><i>N</i> × <i>u</i></span> and the rate with which the whole population goes from all <i>B</i> to all <i>A</i> is the rate that a single mutant <i>A</i> arises times the probability that it will take over the population (<i>fixation probability</i>):

</p>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R=N\cdot u\cdot \rho =u\quad {\text{if}}\quad \rho ={\frac {1}{N}}.}</annotation>

</semantics>

</math></span><img src="./cdb597ae76dc20b675072b4d879ea22ccdd4abef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.095ex; height:5.176ex;" alt="{\displaystyle R=N\cdot u\cdot \rho =u\quad {\text{if}}\quad \rho ={\frac {1}{N}}.}" loading="lazy"></span></dd></dl>

<p>Thus if the mutation is neutral (i.e. the <i>fixation probability</i> is just 1/<i>N</i>) then the rate with which an allele arises and takes over a population is independent of the population size and is equal to the mutation rate. This important result is the basis of the <a href="Neutral_theory_of_evolution" class="mw-redirect" title="Neutral theory of evolution">neutral theory of evolution</a> and suggests that the number of observed point mutations in the <a href="Genome" title="Genome">genomes</a> of two different <a href="Species" title="Species">species</a> would simply be given by the mutation rate multiplied by two times the time since <a href="Divergence" title="Divergence">divergence</a>. Thus the neutral theory of evolution provides a <a href="Molecular_clock" title="Molecular clock">molecular clock</a>, given that the assumptions are fulfilled which may not be the case in reality.

</p>

<ul><li><a href="Weak_Selection" class="mw-redirect" title="Weak Selection">Weak Selection</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFMoran1958" class="citation journal cs1"><a href="Pat_Moran_(statistician)" class="mw-redirect" title="Pat Moran (statistician)">Moran, P. A. P.</a> (1958). "Random processes in genetics". <i><a href="Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society" title="Mathematical Proceedings of the Cambridge Philosophical Society">Mathematical Proceedings of the Cambridge Philosophical Society</a></i>. <b>54</b> (1): <span class="nowrap">60–</span>71. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1958PCPS...54...60M">1958PCPS...54...60M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0305004100033193">10.1017/S0305004100033193</a>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFNowak2006" class="citation book cs1">Nowak, Martin A. (2006). <i>Evolutionary Dynamics: Exploring the Equations of Life</i>. Belknap Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-674-02338-3</bdi>.</cite></li>

<li><cite id="CITEREFMoran1962" class="citation book cs1">Moran, Patrick Alfred Pierce (1962). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/statisticalproce0000mora"><i>The Statistical Processes of Evolutionary Theory</i></a></span>. Oxford: Clarendon Press.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.univie.ac.at/virtuallabs/Moran/">"Evolutionary Dynamics on Graphs"</a>.</cite></li></ul>

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</style><div id="Stochastic_processes496" style="font-size:114%;margin:0 4em"><a href="Stochastic_process" title="Stochastic process">Stochastic processes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Discrete-time_stochastic_process" class="mw-redirect" title="Discrete-time stochastic process">Discrete time</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li>

<li><a href="Branching_process" title="Branching process">Branching process</a></li>

<li><a href="Chinese_restaurant_process" title="Chinese restaurant process">Chinese restaurant process</a></li>

<li><a href="Galton%E2%80%93Watson_process" title="Galton–Watson process">Galton–Watson process</a></li>

<li><a href="Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">Independent and identically distributed random variables</a></li>

<li><a href="Markov_chain" title="Markov chain">Markov chain</a></li>

<li><a href="Random_walk" title="Random walk">Random walk</a>

<ul><li><a href="Loop-erased_random_walk" title="Loop-erased random walk">Loop-erased</a></li>

<li><a href="Self-avoiding_walk" title="Self-avoiding walk">Self-avoiding</a></li>

<li><a href="Biased_random_walk_on_a_graph" title="Biased random walk on a graph"> Biased</a></li>

<li><a href="Maximal_entropy_random_walk" title="Maximal entropy random walk">Maximal entropy</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Continuous-time_stochastic_process" title="Continuous-time stochastic process">Continuous time</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Additive_process" title="Additive process">Additive process</a></li>

<li><a href="Airy_process" title="Airy process">Airy process</a></li>

<li><a href="Bessel_process" title="Bessel process">Bessel process</a></li>

<li><a href="Birth%E2%80%93death_process" title="Birth–death process">Birth–death process</a>

<ul><li><a href="Birth_process" title="Birth process">pure birth</a></li></ul></li>

<li><a href="Wiener_process" title="Wiener process">Brownian motion</a>

<ul><li><a href="Brownian_bridge" title="Brownian bridge">Bridge</a></li>

<li><a href="Dyson_Brownian_motion" title="Dyson Brownian motion">Dyson</a></li>

<li><a href="Brownian_excursion" title="Brownian excursion">Excursion</a></li>

<li><a href="Fractional_Brownian_motion" title="Fractional Brownian motion">Fractional</a></li>

<li><a href="Geometric_Brownian_motion" title="Geometric Brownian motion">Geometric</a></li>

<li><a href="Brownian_meander" title="Brownian meander">Meander</a></li></ul></li>

<li><a href="Cauchy_process" title="Cauchy process">Cauchy process</a></li>

<li><a href="Contact_process_(mathematics)" title="Contact process (mathematics)">Contact process</a></li>

<li><a href="Continuous-time_random_walk" title="Continuous-time random walk">Continuous-time random walk</a></li>

<li><a href="Cox_process" title="Cox process">Cox process</a></li>

<li><a href="Diffusion_process" title="Diffusion process">Diffusion process</a></li>

<li><a href="Empirical_process" title="Empirical process">Empirical process</a></li>

<li><a href="Feller_process" title="Feller process">Feller process</a></li>

<li><a href="Fleming%E2%80%93Viot_process" title="Fleming–Viot process">Fleming–Viot process</a></li>

<li><a href="Gamma_process" title="Gamma process">Gamma process</a></li>

<li><a href="Geometric_process" title="Geometric process">Geometric process</a></li>

<li><a href="Hawkes_process" title="Hawkes process">Hawkes process</a></li>

<li><a href="Hunt_process" title="Hunt process">Hunt process</a></li>

<li><a href="Interacting_particle_system" title="Interacting particle system">Interacting particle systems</a></li>

<li><a href="It%C3%B4_diffusion" title="Itô diffusion">Itô diffusion</a></li>

<li><a href="It%C3%B4_process" class="mw-redirect" title="Itô process">Itô process</a></li>

<li><a href="Jump_diffusion" title="Jump diffusion">Jump diffusion</a></li>

<li><a href="Jump_process" title="Jump process">Jump process</a></li>

<li><a href="L%C3%A9vy_process" title="Lévy process">Lévy process</a></li>

<li><a href="Local_time_(mathematics)" title="Local time (mathematics)">Local time</a></li>

<li><a href="Markov_additive_process" title="Markov additive process">Markov additive process</a></li>

<li><a href="McKean%E2%80%93Vlasov_process" title="McKean–Vlasov process">McKean–Vlasov process</a></li>

<li><a href="Ornstein%E2%80%93Uhlenbeck_process" title="Ornstein–Uhlenbeck process">Ornstein–Uhlenbeck process</a></li>

<li><a href="Poisson_point_process" title="Poisson point process">Poisson process</a>

<ul><li><a href="Compound_Poisson_process" title="Compound Poisson process">Compound</a></li>

<li><a href="Non-homogeneous_Poisson_process" class="mw-redirect" title="Non-homogeneous Poisson process">Non-homogeneous</a></li></ul></li>

<li><a href="Quasimartingale" title="Quasimartingale">Quasimartingale</a></li>

<li><a href="Schramm%E2%80%93Loewner_evolution" title="Schramm–Loewner evolution">Schramm–Loewner evolution</a></li>

<li><a href="Semimartingale" title="Semimartingale">Semimartingale</a></li>

<li><a href="Sigma-martingale" title="Sigma-martingale">Sigma-martingale</a></li>

<li><a href="Stable_process" title="Stable process">Stable process</a></li>

<li><a href="Superprocess" title="Superprocess">Superprocess</a></li>

<li><a href="Telegraph_process" title="Telegraph process">Telegraph process</a></li>

<li><a href="Variance_gamma_process" title="Variance gamma process">Variance gamma process</a></li>

<li><a href="Wiener_process" title="Wiener process">Wiener process</a></li>

<li><a href="Wiener_sausage" title="Wiener sausage">Wiener sausage</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Both</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Branching_process" title="Branching process">Branching process</a></li>

<li><a href="Gaussian_process" title="Gaussian process">Gaussian process</a></li>

<li><a href="Hidden_Markov_model" title="Hidden Markov model">Hidden Markov model (HMM)</a></li>

<li><a href="Markov_process" class="mw-redirect" title="Markov process">Markov process</a></li>

<li><a href="Martingale_(probability_theory)" title="Martingale (probability theory)">Martingale</a>

<ul><li><a href="Martingale_difference_sequence" title="Martingale difference sequence">Differences</a></li>

<li><a href="Local_martingale" title="Local martingale">Local</a></li>

<li><a href="Supermartingale" class="mw-redirect" title="Supermartingale">Super-</a></li></ul></li>

<li><a href="Random_dynamical_system" title="Random dynamical system">Random dynamical system</a></li>

<li><a href="Regenerative_process" title="Regenerative process">Regenerative process</a></li>

<li><a href="Renewal_process" class="mw-redirect" title="Renewal process">Renewal process</a></li>

<li><a href="Stochastic_chains_with_memory_of_variable_length" title="Stochastic chains with memory of variable length">Stochastic chains with memory of variable length</a></li>

<li><a href="White_noise" title="White noise">White noise</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fields and other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dirichlet_process" title="Dirichlet process">Dirichlet process</a></li>

<li><a href="Gaussian_random_field" title="Gaussian random field">Gaussian random field</a></li>

<li><a href="Gibbs_measure" title="Gibbs measure">Gibbs measure</a></li>

<li><a href="Hopfield_model" class="mw-redirect" title="Hopfield model">Hopfield model</a></li>

<li><a href="Ising_model" title="Ising model">Ising model</a>

<ul><li><a href="Potts_model" title="Potts model">Potts model</a></li>

<li><a href="Boolean_network" title="Boolean network">Boolean network</a></li></ul></li>

<li><a href="Markov_random_field" title="Markov random field">Markov random field</a></li>

<li><a href="Percolation_theory" title="Percolation theory">Percolation</a></li>

<li><a href="Pitman%E2%80%93Yor_process" title="Pitman–Yor process">Pitman–Yor process</a></li>

<li><a href="Point_process" title="Point process">Point process</a>

<li><a href="Determinantal_point_process" title="Determinantal point process">Determinantal</a></li>

<li><a href="Poisson_point_process" title="Poisson point process">Poisson</a></li></ul></li>

<li><a href="Random_field" title="Random field">Random field</a></li>

<li><a href="Random_graph" title="Random graph">Random graph</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Time_series" title="Time series">Time series models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH) model</a></li>

<li><a href="Autoregressive_integrated_moving_average" title="Autoregressive integrated moving average">Autoregressive integrated moving average (ARIMA) model</a></li>

<li><a href="Autoregressive_model" title="Autoregressive model">Autoregressive (AR) model</a></li>

<li><a href="Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">Autoregressive–moving-average (ARMA) model</a></li>

<li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Generalized autoregressive conditional heteroskedasticity (GARCH) model</a></li>

<li><a href="Moving-average_model" title="Moving-average model">Moving-average (MA) model</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Asset_pricing_model" class="mw-redirect" title="Asset pricing model">Financial models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Binomial_options_pricing_model" title="Binomial options pricing model">Binomial options pricing model</a></li>

<li><a href="Black%E2%80%93Derman%E2%80%93Toy_model" title="Black–Derman–Toy model">Black–Derman–Toy</a></li>

<li><a href="Black%E2%80%93Karasinski_model" title="Black–Karasinski model">Black–Karasinski</a></li>

<li><a href="Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes</a></li>

<li><a href="Chan%E2%80%93Karolyi%E2%80%93Longstaff%E2%80%93Sanders_process" title="Chan–Karolyi–Longstaff–Sanders process">Chan–Karolyi–Longstaff–Sanders (CKLS)</a></li>

<li><a href="Constant_elasticity_of_variance_model" title="Constant elasticity of variance model">Constant elasticity of variance (CEV)</a></li>

<li><a href="Cox%E2%80%93Ingersoll%E2%80%93Ross_model" title="Cox–Ingersoll–Ross model">Cox–Ingersoll–Ross (CIR)</a></li>

<li><a href="Garman%E2%80%93Kohlhagen_model" class="mw-redirect" title="Garman–Kohlhagen model">Garman–Kohlhagen</a></li>

<li><a href="Heath%E2%80%93Jarrow%E2%80%93Morton_framework" title="Heath–Jarrow–Morton framework">Heath–Jarrow–Morton (HJM)</a></li>

<li><a href="Heston_model" title="Heston model">Heston</a></li>

<li><a href="Hull%E2%80%93White_model" title="Hull–White model">Hull–White</a></li>

<li><a href="Korn%E2%80%93Kreer%E2%80%93Lenssen_model" title="Korn–Kreer–Lenssen model">Korn-Kreer-Lenssen</a></li>

<li><a href="LIBOR_market_model" title="LIBOR market model">LIBOR market</a></li>

<li><a href="Rendleman%E2%80%93Bartter_model" title="Rendleman–Bartter model">Rendleman–Bartter</a></li>

<li><a href="SABR_volatility_model" title="SABR volatility model">SABR volatility</a></li>

<li><a href="Vasicek_model" title="Vasicek model">Vašíček</a></li>

<li><a href="Wilkie_investment_model" title="Wilkie investment model">Wilkie</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="B%C3%BChlmann_model" title="Bühlmann model">Bühlmann</a></li>

<li><a href="Cram%C3%A9r%E2%80%93Lundberg_model" class="mw-redirect" title="Cramér–Lundberg model">Cramér–Lundberg</a></li>

<li><a href="Risk_process" class="mw-redirect" title="Risk process">Risk process</a></li>

<li><a href="Sparre%E2%80%93Anderson_model" class="mw-redirect" title="Sparre–Anderson model">Sparre–Anderson</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Queueing_model" class="mw-redirect" title="Queueing model">Queueing models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Fluid_queue" title="Fluid queue">Fluid</a></li>

<li><a href="G-network" title="G-network">Generalized queueing network</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="C%C3%A0dl%C3%A0g" title="Càdlàg">Càdlàg paths</a></li>

<li><a href="Continuous_stochastic_process" title="Continuous stochastic process">Continuous</a></li>

<li><a href="Sample-continuous_process" title="Sample-continuous process">Continuous paths</a></li>

<li><a href="Ergodicity" title="Ergodicity">Ergodic</a></li>

<li><a href="Exchangeable_random_variables" title="Exchangeable random variables">Exchangeable</a></li>

<li><a href="Feller-continuous_process" title="Feller-continuous process">Feller-continuous</a></li>

<li><a href="Gauss%E2%80%93Markov_process" title="Gauss–Markov process">Gauss–Markov</a></li>

<li><a href="Markov_property" title="Markov property">Markov</a></li>

<li><a href="Mixing_(mathematics)" title="Mixing (mathematics)">Mixing</a></li>

<li><a href="Piecewise-deterministic_Markov_process" title="Piecewise-deterministic Markov process">Piecewise-deterministic</a></li>

<li><a href="Predictable_process" title="Predictable process">Predictable</a></li>

<li><a href="Progressively_measurable_process" title="Progressively measurable process">Progressively measurable</a></li>

<li><a href="Self-similar_process" title="Self-similar process">Self-similar</a></li>

<li><a href="Stationary_process" title="Stationary process">Stationary</a></li>

<li><a href="Time_reversibility" title="Time reversibility">Time-reversible</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Limit theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li>

<li><a href="Donsker's_theorem" title="Donsker's theorem">Donsker's theorem</a></li>

<li><a href="Doob's_martingale_convergence_theorems" title="Doob's martingale convergence theorems">Doob's martingale convergence theorems</a></li>

<li><a href="Ergodic_theorem" class="mw-redirect" title="Ergodic theorem">Ergodic theorem</a></li>

<li><a href="Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem" title="Fisher–Tippett–Gnedenko theorem">Fisher–Tippett–Gnedenko theorem</a></li>

<li><a href="Large_deviation_principle" class="mw-redirect" title="Large deviation principle">Large deviation principle</a></li>

<li><a href="Law_of_large_numbers" title="Law of large numbers">Law of large numbers (weak/strong)</a></li>

<li><a href="Law_of_the_iterated_logarithm" title="Law of the iterated logarithm">Law of the iterated logarithm</a></li>

<li><a href="Maximal_ergodic_theorem" title="Maximal ergodic theorem">Maximal ergodic theorem</a></li>

<li><a href="Sanov's_theorem" title="Sanov's theorem">Sanov's theorem</a></li>

<li><a href="Zero%E2%80%93one_law" title="Zero–one law">Zero–one laws</a> (<a href="Blumenthal's_zero%E2%80%93one_law" title="Blumenthal's zero–one law">Blumenthal</a>, <a href="Borel%E2%80%93Cantelli_lemma" title="Borel–Cantelli lemma">Borel–Cantelli</a>, <a href="Engelbert%E2%80%93Schmidt_zero%E2%80%93one_law" title="Engelbert–Schmidt zero–one law">Engelbert–Schmidt</a>, <a href="Hewitt%E2%80%93Savage_zero%E2%80%93one_law" title="Hewitt–Savage zero–one law">Hewitt–Savage</a>, <a href="Kolmogorov's_zero%E2%80%93one_law" title="Kolmogorov's zero–one law"> Kolmogorov</a>, <a href="L%C3%A9vy's_zero%E2%80%93one_law" class="mw-redirect" title="Lévy's zero–one law">Lévy</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="List_of_inequalities#Probability_theory_and_statistics" title="List of inequalities">Inequalities</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Burkholder%E2%80%93Davis%E2%80%93Gundy_inequalities" class="mw-redirect" title="Burkholder–Davis–Gundy inequalities">Burkholder–Davis–Gundy</a></li>

<li><a href="Doob's_martingale_inequality" title="Doob's martingale inequality">Doob's martingale</a></li>

<li><a href="Doob's_upcrossing_inequality" class="mw-redirect" title="Doob's upcrossing inequality">Doob's upcrossing</a></li>

<li><a href="Kunita%E2%80%93Watanabe_inequality" title="Kunita–Watanabe inequality">Kunita–Watanabe</a></li>

<li><a href="Marcinkiewicz%E2%80%93Zygmund_inequality" title="Marcinkiewicz–Zygmund inequality">Marcinkiewicz–Zygmund</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cameron%E2%80%93Martin_formula" class="mw-redirect" title="Cameron–Martin formula">Cameron–Martin formula</a></li>

<li><a href="Convergence_of_random_variables" title="Convergence of random variables">Convergence of random variables</a></li>

<li><a href="Dol%C3%A9ans-Dade_exponential" title="Doléans-Dade exponential">Doléans-Dade exponential</a></li>

<li><a href="Doob_decomposition_theorem" title="Doob decomposition theorem">Doob decomposition theorem</a></li>

<li><a href="Doob%E2%80%93Meyer_decomposition_theorem" title="Doob–Meyer decomposition theorem">Doob–Meyer decomposition theorem</a></li>

<li><a href="Doob's_optional_stopping_theorem" class="mw-redirect" title="Doob's optional stopping theorem">Doob's optional stopping theorem</a></li>

<li><a href="Dynkin's_formula" title="Dynkin's formula">Dynkin's formula</a></li>

<li><a href="Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a></li>

<li><a href="Filtration_(probability_theory)" title="Filtration (probability theory)">Filtration</a></li>

<li><a href="Girsanov_theorem" title="Girsanov theorem">Girsanov theorem</a></li>

<li><a href="Infinitesimal_generator_(stochastic_processes)" title="Infinitesimal generator (stochastic processes)">Infinitesimal generator</a></li>

<li><a href="It%C3%B4_integral" class="mw-redirect" title="Itô integral">Itô integral</a></li>

<li><a href="It%C3%B4's_lemma" title="Itô's lemma">Itô's lemma</a></li>

<li><a href="Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">Karhunen–Loève theorem</a></li>

<li><a href="Kolmogorov_continuity_theorem" title="Kolmogorov continuity theorem">Kolmogorov continuity theorem</a></li>

<li><a href="Kolmogorov_extension_theorem" title="Kolmogorov extension theorem">Kolmogorov extension theorem</a></li>

<li><a href="L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li>

<li><a href="Malliavin_calculus" title="Malliavin calculus">Malliavin calculus</a></li>

<li><a href="Martingale_representation_theorem" title="Martingale representation theorem">Martingale representation theorem</a></li>

<li><a href="Optional_stopping_theorem" title="Optional stopping theorem">Optional stopping theorem</a></li>

<li><a href="Prokhorov's_theorem" title="Prokhorov's theorem">Prokhorov's theorem</a></li>

<li><a href="Quadratic_variation" title="Quadratic variation">Quadratic variation</a></li>

<li><a href="Reflection_principle_(Wiener_process)" title="Reflection principle (Wiener process)">Reflection principle</a></li>

<li><a href="Skorokhod_integral" title="Skorokhod integral">Skorokhod integral</a></li>

<li><a href="Skorokhod's_representation_theorem" title="Skorokhod's representation theorem">Skorokhod's representation theorem</a></li>

<li><a href="Skorokhod_space" class="mw-redirect" title="Skorokhod space">Skorokhod space</a></li>

<li><a href="Snell_envelope" title="Snell envelope">Snell envelope</a></li>

<li><a href="Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a>

<ul><li><a href="Tanaka_equation" title="Tanaka equation">Tanaka</a></li></ul></li>

<li><a href="Stopping_time" title="Stopping time">Stopping time</a></li>

<li><a href="Stratonovich_integral" title="Stratonovich integral">Stratonovich integral</a></li>

<li><a href="Uniform_integrability" title="Uniform integrability">Uniform integrability</a></li>

<li><a href="Usual_hypotheses" class="mw-redirect" title="Usual hypotheses">Usual hypotheses</a></li>

<li><a href="Wiener_space" class="mw-redirect" title="Wiener space">Wiener space</a>

<ul><li><a href="Classical_Wiener_space" title="Classical Wiener space">Classical</a></li>

<li><a href="Abstract_Wiener_space" title="Abstract Wiener space">Abstract</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Disciplines</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial mathematics</a></li>

<li><a href="Stochastic_control" title="Stochastic control">Control theory</a></li>

<li><a href="Econometrics" title="Econometrics">Econometrics</a></li>

<li><a href="Ergodic_theory" title="Ergodic theory">Ergodic theory</a></li>

<li><a href="Extreme_value_theory" title="Extreme value theory">Extreme value theory (EVT)</a></li>

<li><a href="Large_deviations_theory" title="Large deviations theory">Large deviations theory</a></li>

<li><a href="Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li>

<li><a href="Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li>

<li><a href="Probability_theory" title="Probability theory">Probability theory</a></li>

<li><a href="Queueing_theory" title="Queueing theory">Queueing theory</a></li>

<li><a href="Renewal_theory" title="Renewal theory">Renewal theory</a></li>

<li><a href="Ruin_theory" title="Ruin theory">Ruin theory</a></li>

<li><a href="Signal_processing" title="Signal processing">Signal processing</a></li>

<li><a href="Statistics" title="Statistics">Statistics</a></li>

<li><a href="Stochastic_analysis" class="mw-redirect" title="Stochastic analysis">Stochastic analysis</a></li>

<li><a href="Time_series_analysis" class="mw-redirect" title="Time series analysis">Time series analysis</a></li>

<li><a href="Machine_learning" title="Machine learning">Machine learning</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div>

<ul><li><a href="List_of_stochastic_processes_topics" title="List of stochastic processes topics">List of topics</a></li>

<li>Category</li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-02-23" href="https://en.wikipedia.org/wiki/?title=Moran_process&oldid=1277295443">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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