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<title>Conditional probability distribution</title>

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Conditional probability distribution

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In <a href="Probability_theory" title="Probability theory">probability theory</a> and <a href="Statistics" title="Statistics">statistics</a>, the conditional probability distribution is a probability distribution that describes the probability of an outcome given the occurrence of a particular event. Given two <a href="Joint_probability_distribution" title="Joint probability distribution">jointly distributed</a> <a href="Random_variable" title="Random variable">random variables</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">, the conditional probability distribution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is the <a href="Probability_distribution" title="Probability distribution">probability distribution</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> as a parameter. When both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> are <a href="Categorical_variable" title="Categorical variable">categorical variables</a>, a <a href="Conditional_probability_table" title="Conditional probability table">conditional probability table</a> is typically used to represent the conditional probability. The conditional distribution contrasts with the <a href="Marginal_distribution" title="Marginal distribution">marginal distribution</a> of a random variable, which is its distribution without reference to the value of the other variable.

If the conditional distribution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a <a href="Continuous_distribution" class="mw-redirect" title="Continuous distribution">continuous distribution</a>, then its <a href="Probability_density_function" title="Probability density function">probability density function</a> is known as the conditional density function.<a href="#cite_note-FOOTNOTERoss199388–91-1">[1]</a> The properties of a conditional distribution, such as the <a href="Moment_(mathematics)" title="Moment (mathematics)">moments</a>, are often referred to by corresponding names such as the <a href="Conditional_mean" class="mw-redirect" title="Conditional mean">conditional mean</a> and <a href="Conditional_variance" title="Conditional variance">conditional variance</a>.

More generally, one can refer to the conditional distribution of a subset of a set of more than two variables; this conditional distribution is contingent on the values of all the remaining variables, and if more than one variable is included in the subset then this conditional distribution is the conditional <a href="Joint_distribution" class="mw-redirect" title="Joint distribution">joint distribution</a> of the included variables.

<div class="mw-heading mw-heading2"><h2 id="Conditional_discrete_distributions">Conditional discrete distributions</h2></div>

For <a href="Discrete_random_variable" class="mw-redirect" title="Discrete random variable">discrete random variables</a>, the conditional <a href="Probability_mass_function" title="Probability mass function">probability mass function</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=x}">

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

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</math><img src="./0661396d873679039ffe8e908a39f02402d4912d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.408ex; height:2.176ex;" alt="{\displaystyle X=x}" loading="lazy"> can be written according to its definition as:

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<annotation encoding="application/x-tex">{\displaystyle p_{Y|X}(y\mid x)\triangleq P(Y=y\mid X=x)={\frac {P(\{X=x\}\cap \{Y=y\})}{P(X=x)}}\qquad }</annotation>

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</math><img src="./2e7b939db74fbc9d7157d5be719e12de60d73c3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.089ex; width:63.662ex; height:6.509ex;" alt="{\displaystyle p_{Y|X}(y\mid x)\triangleq P(Y=y\mid X=x)={\frac {P(\{X=x\}\cap \{Y=y\})}{P(X=x)}}\qquad }" loading="lazy"></dd></dl>

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Due to the occurrence of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X=x)}">

</mstyle>

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

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</math><img src="./073aa90e978ecf7072bb4afcf04ba2ea00140803.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.963ex; height:2.843ex;" alt="{\displaystyle P(X=x)}" loading="lazy"> in the denominator, this is defined only for non-zero (hence strictly positive) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X=x).}">

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

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</math><img src="./e525a0719c5bfacd822ed6f9c05999934580e224.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.61ex; height:2.843ex;" alt="{\displaystyle P(X=x).}" loading="lazy">

The relation with the probability distribution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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</mrow>

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

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is:

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<annotation encoding="application/x-tex">{\displaystyle P(Y=y\mid X=x)P(X=x)=P(\{X=x\}\cap \{Y=y\})=P(X=x\mid Y=y)P(Y=y).}</annotation>

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</math><img src="./c40a8b4cd88292f2243eb9766deb058494809a6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:85.466ex; height:2.843ex;" alt="{\displaystyle P(Y=y\mid X=x)P(X=x)=P(\{X=x\}\cap \{Y=y\})=P(X=x\mid Y=y)P(Y=y).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Example">Example</h3></div>

Consider the roll of a fair die and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=1}">

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

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</math><img src="./889527b2a786390a016fc3ef7cd8eee77e86b6f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.241ex; height:2.176ex;" alt="{\displaystyle X=1}" loading="lazy"> if the number is even (i.e., 2, 4, or 6) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=0}">

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

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</math><img src="./d519e9e94f279ea82581dfa70a2444e896e2d860.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.241ex; height:2.176ex;" alt="{\displaystyle X=0}" loading="lazy"> otherwise. Furthermore, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=1}">

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

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</math><img src="./867ae2de7c84119e258e68ca484e01e03b00bd73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.034ex; height:2.176ex;" alt="{\displaystyle Y=1}" loading="lazy"> if the number is prime (i.e., 2, 3, or 5) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=0}">

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</mrow>

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

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</math><img src="./56cd853e6606465d2259975da9d0a0bb08f612af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.034ex; height:2.176ex;" alt="{\displaystyle Y=0}" loading="lazy"> otherwise.

<th>6

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<td>1

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<td>0

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Then the unconditional probability that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=1}">

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

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</math><img src="./889527b2a786390a016fc3ef7cd8eee77e86b6f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.241ex; height:2.176ex;" alt="{\displaystyle X=1}" loading="lazy"> is 3/6 = 1/2 (since there are six possible rolls of the dice, of which three are even), whereas the probability that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=1}">

</mstyle>

</mrow>

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

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</math><img src="./889527b2a786390a016fc3ef7cd8eee77e86b6f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.241ex; height:2.176ex;" alt="{\displaystyle X=1}" loading="lazy"> conditional on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./867ae2de7c84119e258e68ca484e01e03b00bd73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.034ex; height:2.176ex;" alt="{\displaystyle Y=1}" loading="lazy"> is 1/3 (since there are three possible <a href="Prime_number" title="Prime number">prime number</a> rolls—2, 3, and 5—of which one is even).

<div class="mw-heading mw-heading2"><h2 id="Conditional_continuous_distributions">Conditional continuous distributions</h2></div>

Similarly for <a href="Continuous_random_variable" class="mw-redirect" title="Continuous random variable">continuous random variables</a>, the conditional <a href="Probability_density_function" title="Probability density function">probability density function</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> given the occurrence of the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> can be written as<a href="#cite_note-FOOTNOTEPark201899-2">[2]</a>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle f_{Y\mid X}(y\mid x)={\frac {f_{X,Y}(x,y)}{f_{X}(x)}}\qquad }</annotation>

</semantics>

</math><img src="./8629d6369aa7ac1371fc44ee90bf4c9099b7fb08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.106ex; height:6.509ex;" alt="{\displaystyle f_{Y\mid X}(y\mid x)={\frac {f_{X,Y}(x,y)}{f_{X}(x)}}\qquad }" loading="lazy"></dd></dl>

</div>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X,Y}(x,y)}">

<msub>

</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle f_{X,Y}(x,y)}</annotation>

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</math><img src="./f2e305446e0fee9c51aee21bc81fb7136f1151e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.811ex; height:3.009ex;" alt="{\displaystyle f_{X,Y}(x,y)}" loading="lazy"> gives the <a href="Joint_distribution" class="mw-redirect" title="Joint distribution">joint density</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">, while <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

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</math><img src="./29310a010e7f7dfa33ba69bcf1ef9ec166d461dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.911ex; height:2.843ex;" alt="{\displaystyle f_{X}(x)}" loading="lazy"> gives the <a href="Marginal_density" class="mw-redirect" title="Marginal density">marginal density</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. Also in this case it is necessary that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}(x)>0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b0a59e1d2780218984d906bee9f1c73c18f3f56f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.171ex; height:2.843ex;" alt="{\displaystyle f_{X}(x)>0}" loading="lazy">.

The relation with the probability distribution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is given by:

<msub>

</mrow>

</msub>

<msub>

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<msub>

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</msub>

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle f_{Y\mid X}(y\mid x)f_{X}(x)=f_{X,Y}(x,y)=f_{X|Y}(x\mid y)f_{Y}(y).}</annotation>

</semantics>

</math><img src="./d14fd769cac09c5cde6df82029e88e4ad0d7ebc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:49.585ex; height:3.176ex;" alt="{\displaystyle f_{Y\mid X}(y\mid x)f_{X}(x)=f_{X,Y}(x,y)=f_{X|Y}(x\mid y)f_{Y}(y).}" loading="lazy"></dd></dl>

The concept of the conditional distribution of a continuous random variable is not as intuitive as it might seem: <a href="Borel's_paradox" class="mw-redirect" title="Borel's paradox">Borel's paradox</a> shows that conditional probability density functions need not be invariant under coordinate transformations.

<div class="mw-heading mw-heading3"><h3 id="Example_2">Example</h3></div>

The graph shows a <a href="Bivariate_normal_distribution" class="mw-redirect" title="Bivariate normal distribution">bivariate normal joint density</a> for random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">. To see the distribution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> conditional on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=70}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=70}</annotation>

</semantics>

</math><img src="./b5f2453df554b44fb22a284f0ef02aa6a100a487.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.403ex; height:2.176ex;" alt="{\displaystyle X=70}" loading="lazy">, one can first visualize the line <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=70}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=70}</annotation>

</semantics>

</math><img src="./b5f2453df554b44fb22a284f0ef02aa6a100a487.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.403ex; height:2.176ex;" alt="{\displaystyle X=70}" loading="lazy"> in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,Y}</annotation>

</semantics>

</math><img src="./b8705438171d938b7f59cd1bfa5b7d99b6afa5cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.787ex; height:2.509ex;" alt="{\displaystyle X,Y}" loading="lazy"> <a href="Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a>, and then visualize the plane containing that line and perpendicular to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,Y}</annotation>

</semantics>

</math><img src="./b8705438171d938b7f59cd1bfa5b7d99b6afa5cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.787ex; height:2.509ex;" alt="{\displaystyle X,Y}" loading="lazy"> plane. The intersection of that plane with the joint normal density, once rescaled to give unit area under the intersection, is the relevant conditional density of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">.

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

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<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y\mid X=70\ \sim \ {\mathcal {N}}\left(\mu _{Y}+{\frac {\sigma _{Y}}{\sigma _{X}}}\rho (70-\mu _{X}),\,(1-\rho ^{2})\sigma _{Y}^{2}\right).}</annotation>

</semantics>

</math><img src="./e9cefdef94f07d364bee1e42d61144d98094f98f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:55.528ex; height:6.176ex;" alt="{\displaystyle Y\mid X=70\ \sim \ {\mathcal {N}}\left(\mu _{Y}+{\frac {\sigma _{Y}}{\sigma _{X}}}\rho (70-\mu _{X}),\,(1-\rho ^{2})\sigma _{Y}^{2}\right).}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Relation_to_independence">Relation to independence</h2></div>

Random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> are <a href="Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a> <a href="If_and_only_if" title="If and only if">if and only if</a> the conditional distribution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is, for all possible realizations of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, equal to the unconditional distribution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">. For discrete random variables this means <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(Y=y|X=x)=P(Y=y)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(Y=y|X=x)=P(Y=y)}</annotation>

</semantics>

</math><img src="./47f7d6b22cbbc1b3f13ff3c4855294b3654499e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.317ex; height:2.843ex;" alt="{\displaystyle P(Y=y|X=x)=P(Y=y)}" loading="lazy"> for all possible <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X=x)>0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e865b191d2915dac79312558c5fc49d0d9d8d84a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.224ex; height:2.843ex;" alt="{\displaystyle P(X=x)>0}" loading="lazy">. For continuous random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">, having a <a href="Joint_density_function" class="mw-redirect" title="Joint density function">joint density function</a>, it means <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{Y}(y|X=x)=f_{Y}(y)}">

<msub>

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</msub>

</mrow>

<msub>

</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle f_{Y}(y|X=x)=f_{Y}(y)}</annotation>

</semantics>

</math><img src="./9d6313de51cf3bd0751279695956d2bf6eb42a88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.334ex; height:2.843ex;" alt="{\displaystyle f_{Y}(y|X=x)=f_{Y}(y)}" loading="lazy"> for all possible <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}(x)>0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b0a59e1d2780218984d906bee9f1c73c18f3f56f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.171ex; height:2.843ex;" alt="{\displaystyle f_{X}(x)>0}" loading="lazy">.

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

Seen as a function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> for given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(Y=y|X=x)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(Y=y|X=x)}</annotation>

</semantics>

</math><img src="./b158f15d6b4cb725f0024d7d9e31dce5c1691dba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.637ex; height:2.843ex;" alt="{\displaystyle P(Y=y|X=x)}" loading="lazy"> is a probability mass function and so the sum over all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> (or integral if it is a conditional probability density) is 1. Seen as a function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> for given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">, it is a <a href="Likelihood_function" title="Likelihood function">likelihood function</a>, so that the sum (or integral) over all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> need not be 1.

Additionally, a marginal of a joint distribution can be expressed as the expectation of the corresponding conditional distribution. For instance, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{X}(x)=E_{Y}[p_{X|Y}(x\ |\ Y)]}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{X}(x)=E_{Y}[p_{X|Y}(x\ |\ Y)]}</annotation>

</semantics>

</math><img src="./776889312ebcbef1c2662de91d222f99f3e814e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-left: -0.089ex; width:24.857ex; height:3.176ex;" alt="{\displaystyle p_{X}(x)=E_{Y}[p_{X|Y}(x\ |\ Y)]}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Measure-theoretic_formulation">Measure-theoretic formulation</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},P)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},P)}</annotation>

</semantics>

</math><img src="./9d77104a5c3c49cc0634dcf6908db7ad45f738d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},P)}" loading="lazy"> be a <a href="Probability_space" title="Probability space">probability space</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}\subseteq {\mathcal {F}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}\subseteq {\mathcal {F}}}</annotation>

</semantics>

</math><img src="./48de993de3b2712e340de231c288c277880a155c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.417ex; height:2.343ex;" alt="{\displaystyle {\mathcal {G}}\subseteq {\mathcal {F}}}" loading="lazy"> a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">-field in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy">. Given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in {\mathcal {F}}}</annotation>

</semantics>

</math><img src="./8cdf27fd56b3c06d1ddf9ad1efd7cee0a81cb2dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.51ex; height:2.176ex;" alt="{\displaystyle A\in {\mathcal {F}}}" loading="lazy">, the <a href="Radon%E2%80%93Nikodym_theorem" title="Radon–Nikodym theorem">Radon–Nikodym theorem</a> implies that there is<a href="#cite_note-FOOTNOTEBillingsley1995430-3">[3]</a> a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a980c59d42c003fd07fdf3646e1fb95ff82f99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.392ex; height:2.343ex;" alt="{\displaystyle {\mathcal {G}}}" loading="lazy">-measurable random variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A\mid {\mathcal {G}}):\Omega \to \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(A\mid {\mathcal {G}}):\Omega \to \mathbb {R} }</annotation>

</semantics>

</math><img src="./66979e935a4c5ac88228deb389d587cc5088dea8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.535ex; height:2.843ex;" alt="{\displaystyle P(A\mid {\mathcal {G}}):\Omega \to \mathbb {R} }" loading="lazy">, called the <a href="Conditional_probability" title="Conditional probability">conditional probability</a>, such that<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{G}P(A\mid {\mathcal {G}})(\omega )dP(\omega )=P(A\cap G)}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{G}P(A\mid {\mathcal {G}})(\omega )dP(\omega )=P(A\cap G)}</annotation>

</semantics>

</math>for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\in {\mathcal {G}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\in {\mathcal {G}}}</annotation>

</semantics>

</math><img src="./c125d7ec0f60185da3561672bb09a38074a1a6d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.06ex; height:2.343ex;" alt="{\displaystyle G\in {\mathcal {G}}}" loading="lazy">, and such a random variable is uniquely defined up to sets of probability zero. A conditional probability is called <a href="Regular_conditional_probability" title="Regular conditional probability">regular</a> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {P} (\cdot \mid {\mathcal {G}})(\omega )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (\cdot \mid {\mathcal {G}})(\omega )}</annotation>

</semantics>

</math><img src="./beefc80369e7ae1d0106ae3df2233b75aaa0fd3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.624ex; height:2.843ex;" alt="{\displaystyle \operatorname {P} (\cdot \mid {\mathcal {G}})(\omega )}" loading="lazy"> is a <a href="Probability_measure" title="Probability measure">probability measure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}})}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}})}</annotation>

</semantics>

</math><img src="./4af05190f6fb10d1eec88b5f66f940dc88b5ec87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.448ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}})}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \in \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega \in \Omega }</annotation>

</semantics>

</math><img src="./e23a8931e1b954519d9fb9ba2e7f02eaa11ac91a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.965ex; height:2.176ex;" alt="{\displaystyle \omega \in \Omega }" loading="lazy"> a.e.

Special cases:

<ul><li>For the trivial sigma algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}=\{\emptyset ,\Omega \}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}=\{\emptyset ,\Omega \}}</annotation>

</semantics>

</math><img src="./90bd37a752211b48ca9b341b52c0fa6fc5b7af02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.69ex; height:2.843ex;" alt="{\displaystyle {\mathcal {G}}=\{\emptyset ,\Omega \}}" loading="lazy">, the conditional probability is the constant function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {P} \!\left(A\mid \{\emptyset ,\Omega \}\right)=\operatorname {P} (A).}">

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {P} \!\left(A\mid \{\emptyset ,\Omega \}\right)=\operatorname {P} (A).}</annotation>

</semantics>

</math><img src="./08a209e83449217e366c2c0884cc8814de1af257.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.152ex; height:2.843ex;" alt="{\displaystyle \operatorname {P} \!\left(A\mid \{\emptyset ,\Omega \}\right)=\operatorname {P} (A).}" loading="lazy"></li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in {\mathcal {G}}}</annotation>

</semantics>

</math><img src="./6aa279df8d3065fab30c4a5f57b15b996ee71748.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.976ex; height:2.343ex;" alt="{\displaystyle A\in {\mathcal {G}}}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {P} (A\mid {\mathcal {G}})=1_{A}}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (A\mid {\mathcal {G}})=1_{A}}</annotation>

</semantics>

</math><img src="./657f1fb96a3d5022e6eebdd9929cac30f33f3b9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.191ex; height:2.843ex;" alt="{\displaystyle \operatorname {P} (A\mid {\mathcal {G}})=1_{A}}" loading="lazy">, the indicator function (defined <a href="#Relation_to_conditional_expectation">below</a>).</li></ul>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X:\Omega \to E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X:\Omega \to E}</annotation>

</semantics>

</math><img src="./fa5b5c95b9df3bce45c8ba6012f7434a075f79f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.985ex; height:2.176ex;" alt="{\displaystyle X:\Omega \to E}" loading="lazy"> be a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,{\mathcal {E}})}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (E,{\mathcal {E}})}</annotation>

</semantics>

</math><img src="./497b309dc3f6f358722b8a00936c8e1ed5c787b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.93ex; height:2.843ex;" alt="{\displaystyle (E,{\mathcal {E}})}" loading="lazy">-valued random variable. For each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\in {\mathcal {E}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\in {\mathcal {E}}}</annotation>

</semantics>

</math><img src="./81710349199c3fc6bcbe14f6758e0c2f1fe9bbca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.916ex; height:2.176ex;" alt="{\displaystyle B\in {\mathcal {E}}}" loading="lazy">, define <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{X\,|\,{\mathcal {G}}}(B\,|\,{\mathcal {G}})=\mathrm {P} (X^{-1}(B)\,|\,{\mathcal {G}}).}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu _{X\,|\,{\mathcal {G}}}(B\,|\,{\mathcal {G}})=\mathrm {P} (X^{-1}(B)\,|\,{\mathcal {G}}).}</annotation>

</semantics>

</math>For any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \in \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega \in \Omega }</annotation>

</semantics>

</math><img src="./e23a8931e1b954519d9fb9ba2e7f02eaa11ac91a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.965ex; height:2.176ex;" alt="{\displaystyle \omega \in \Omega }" loading="lazy">, the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{X\,|{\mathcal {G}}}(\cdot \,|{\mathcal {G}})(\omega ):{\mathcal {E}}\to \mathbb {R} }">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu _{X\,|{\mathcal {G}}}(\cdot \,|{\mathcal {G}})(\omega ):{\mathcal {E}}\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./302b3e3cc1908be57c279d250623f8f7a3847c1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.541ex; height:3.176ex;" alt="{\displaystyle \mu _{X\,|{\mathcal {G}}}(\cdot \,|{\mathcal {G}})(\omega ):{\mathcal {E}}\to \mathbb {R} }" loading="lazy"> is called the conditional probability distribution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a980c59d42c003fd07fdf3646e1fb95ff82f99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.392ex; height:2.343ex;" alt="{\displaystyle {\mathcal {G}}}" loading="lazy">. If it is a probability measure on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,{\mathcal {E}})}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (E,{\mathcal {E}})}</annotation>

</semantics>

</math><img src="./497b309dc3f6f358722b8a00936c8e1ed5c787b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.93ex; height:2.843ex;" alt="{\displaystyle (E,{\mathcal {E}})}" loading="lazy">, then it is called <a href="Regular_conditional_probability" title="Regular conditional probability">regular</a>.

For a real-valued random variable (with respect to the Borel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">-field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}^{1}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./52ca5ddfe5eb21dcfaa0e55bb2d5104973199790.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.025ex; height:2.676ex;" alt="{\displaystyle {\mathcal {R}}^{1}}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">), every conditional probability distribution is regular.<a href="#cite_note-FOOTNOTEBillingsley1995439-4">[4]</a> In this case,<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[X\mid {\mathcal {G}}]=\int _{-\infty }^{\infty }x\,\mu _{X\mid {\mathcal {G}}}(dx,\cdot )}">

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E[X\mid {\mathcal {G}}]=\int _{-\infty }^{\infty }x\,\mu _{X\mid {\mathcal {G}}}(dx,\cdot )}</annotation>

</semantics>

</math><img src="./96b1d6a44a4cb2f0916cba84dc987710700b92aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.539ex; height:6.009ex;" alt="{\displaystyle E[X\mid {\mathcal {G}}]=\int _{-\infty }^{\infty }x\,\mu _{X\mid {\mathcal {G}}}(dx,\cdot )}" loading="lazy"> almost surely.

<div class="mw-heading mw-heading3"><h3 id="Relation_to_conditional_expectation">Relation to conditional expectation</h3></div>

For any event <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in {\mathcal {F}}}</annotation>

</semantics>

</math><img src="./8cdf27fd56b3c06d1ddf9ad1efd7cee0a81cb2dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.51ex; height:2.176ex;" alt="{\displaystyle A\in {\mathcal {F}}}" loading="lazy">, define the <a href="Indicator_function" title="Indicator function">indicator function</a>:

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{A}(\omega )={\begin{cases}1\;&{\text{if }}\omega \in A,\\0\;&{\text{if }}\omega \notin A,\end{cases}}}</annotation>

</semantics>

</math><img src="./de81db883f78aa94b72f8863c61061678013122f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.396ex; height:6.176ex;" alt="{\displaystyle \mathbf {1} _{A}(\omega )={\begin{cases}1\;&{\text{if }}\omega \in A,\\0\;&{\text{if }}\omega \notin A,\end{cases}}}" loading="lazy"></dd></dl>

which is a random variable. Note that the expectation of this random variable is equal to the probability of A itself:

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (\mathbf {1} _{A})=\operatorname {P} (A).\;}</annotation>

</semantics>

</math><img src="./1ca8d3faa02670ed2b0be89aa19a63deec51b73a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.719ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} (\mathbf {1} _{A})=\operatorname {P} (A).\;}" loading="lazy"></dd></dl>

Given a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">-field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}\subseteq {\mathcal {F}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}\subseteq {\mathcal {F}}}</annotation>

</semantics>

</math><img src="./48de993de3b2712e340de231c288c277880a155c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.417ex; height:2.343ex;" alt="{\displaystyle {\mathcal {G}}\subseteq {\mathcal {F}}}" loading="lazy">, the conditional probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {P} (A\mid {\mathcal {G}})}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (A\mid {\mathcal {G}})}</annotation>

</semantics>

</math><img src="./8115319855edb4fa36befd7febfca17a8656b975.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.465ex; height:2.843ex;" alt="{\displaystyle \operatorname {P} (A\mid {\mathcal {G}})}" loading="lazy"> is a version of the <a href="Conditional_expectation" title="Conditional expectation">conditional expectation</a> of the indicator function for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">:

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (A\mid {\mathcal {G}})=\operatorname {E} (\mathbf {1} _{A}\mid {\mathcal {G}})\;}</annotation>

</semantics>

</math><img src="./226ec4d4a1ddfd001e9448950cfd62c998b359f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.732ex; height:2.843ex;" alt="{\displaystyle \operatorname {P} (A\mid {\mathcal {G}})=\operatorname {E} (\mathbf {1} _{A}\mid {\mathcal {G}})\;}" loading="lazy"></dd></dl>

An expectation of a random variable with respect to a regular conditional probability is equal to its conditional expectation.

<div class="mw-heading mw-heading3"><h3 id="Interpretation_of_conditioning_on_a_Sigma_Field">Interpretation of conditioning on a Sigma Field</h3></div>

Consider the probability space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}</annotation>

</semantics>

</math><img src="./6bb8743f7565082ed1a9ee0490d9d71be82eafaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.902ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}" loading="lazy">

and a sub-sigma field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}\subset {\mathcal {F}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}\subset {\mathcal {F}}}</annotation>

</semantics>

</math><img src="./e5f4e9e64f8f32765a88ae5e73c0bf1d84816ecc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.928ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}\subset {\mathcal {F}}}" loading="lazy">.

The sub-sigma field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./280ae03440942ab348c2ca9b8db6b56ffa9618f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" loading="lazy"> can be loosely interpreted as containing a subset of the information in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy">. For example, we might think of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} (B|{\mathcal {A}})}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (B|{\mathcal {A}})}</annotation>

</semantics>

</math><img src="./27d6d3555d501c904ba7ac942391a38742225f73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.544ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (B|{\mathcal {A}})}" loading="lazy"> as the probability of the event <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> given the information in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./280ae03440942ab348c2ca9b8db6b56ffa9618f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" loading="lazy">.

Also recall that an event <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> is independent of a sub-sigma field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./280ae03440942ab348c2ca9b8db6b56ffa9618f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} (B|A)=\mathbb {P} (B)}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (B|A)=\mathbb {P} (B)}</annotation>

</semantics>

</math><img src="./4ddb70faf0204fa8aec74b533485dfa5f154bcec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.476ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (B|A)=\mathbb {P} (B)}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {A}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in {\mathcal {A}}}</annotation>

</semantics>

</math><img src="./d6e69647797be244cf2ebc28ecd61fafba8790c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.487ex; height:2.343ex;" alt="{\displaystyle A\in {\mathcal {A}}}" loading="lazy">. It is incorrect to conclude in general that the information in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./280ae03440942ab348c2ca9b8db6b56ffa9618f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" loading="lazy"> does not tell us anything about the probability of event <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> occurring. This can be shown with a counter-example:

Consider a probability space on the <a href="Unit_interval" title="Unit interval">unit interval</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =[0,1]}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a16b39940ed262a459907035d290bf78c18134b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.429ex; height:2.843ex;" alt="{\displaystyle \Omega =[0,1]}" loading="lazy">. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a980c59d42c003fd07fdf3646e1fb95ff82f99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.392ex; height:2.343ex;" alt="{\displaystyle {\mathcal {G}}}" loading="lazy"> be the sigma-field of all countable sets and sets whose complement is countable. So each set in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a980c59d42c003fd07fdf3646e1fb95ff82f99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.392ex; height:2.343ex;" alt="{\displaystyle {\mathcal {G}}}" loading="lazy"> has measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> and so is independent of each event in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy">. However, notice that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a980c59d42c003fd07fdf3646e1fb95ff82f99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.392ex; height:2.343ex;" alt="{\displaystyle {\mathcal {G}}}" loading="lazy"> also contains all the singleton events in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"> (those sets which contain only a single <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \in \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega \in \Omega }</annotation>

</semantics>

</math><img src="./e23a8931e1b954519d9fb9ba2e7f02eaa11ac91a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.965ex; height:2.176ex;" alt="{\displaystyle \omega \in \Omega }" loading="lazy">). So knowing which of the events in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a980c59d42c003fd07fdf3646e1fb95ff82f99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.392ex; height:2.343ex;" alt="{\displaystyle {\mathcal {G}}}" loading="lazy"> occurred is equivalent to knowing exactly which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \in \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega \in \Omega }</annotation>

</semantics>

</math><img src="./e23a8931e1b954519d9fb9ba2e7f02eaa11ac91a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.965ex; height:2.176ex;" alt="{\displaystyle \omega \in \Omega }" loading="lazy"> occurred! So in one sense, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a980c59d42c003fd07fdf3646e1fb95ff82f99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.392ex; height:2.343ex;" alt="{\displaystyle {\mathcal {G}}}" loading="lazy"> contains no information about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"> (it is independent of it), and in another sense it contains all the information in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy">.<a href="#cite_note-FOOTNOTEBillingsley2012-5">[5]</a>

<ul><li><a href="Conditioning_(probability)" title="Conditioning (probability)">Conditioning (probability)</a></li>

<li><a href="Conditional_probability" title="Conditional probability">Conditional probability</a></li>

<li><a href="Regular_conditional_probability" title="Regular conditional probability">Regular conditional probability</a></li>

<li><a href="Bayes'_theorem" title="Bayes' theorem">Bayes' theorem</a></li></ul>

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

<div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3></div>

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<li id="cite_note-FOOTNOTERoss199388–91-1"><a href="#cite_ref-FOOTNOTERoss199388–91_1-0">^</a> <a href="#CITEREFRoss1993">Ross (1993)</a>, pp. 88–91.

</li>

<li id="cite_note-FOOTNOTEPark201899-2"><a href="#cite_ref-FOOTNOTEPark201899_2-0">^</a> <a href="#CITEREFPark2018">Park (2018)</a>, p. 99.

</li>

<li id="cite_note-FOOTNOTEBillingsley1995430-3"><a href="#cite_ref-FOOTNOTEBillingsley1995430_3-0">^</a> <a href="#CITEREFBillingsley1995">Billingsley (1995)</a>, p. 430.

</li>

<li id="cite_note-FOOTNOTEBillingsley1995439-4"><a href="#cite_ref-FOOTNOTEBillingsley1995439_4-0">^</a> <a href="#CITEREFBillingsley1995">Billingsley (1995)</a>, p. 439.

</li>

<li id="cite_note-FOOTNOTEBillingsley2012-5"><a href="#cite_ref-FOOTNOTEBillingsley2012_5-0">^</a> <a href="#CITEREFBillingsley2012">Billingsley (2012)</a>.

</li>

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

<div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3></div>

/* start https://en.wikipedia.org/ */

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFBillingsley1995" class="citation book cs1"><a href="Patrick_Billingsley" title="Patrick Billingsley">Billingsley, Patrick</a> (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=a3gavZbxyJcC">Probability and Measure</a> (3rd ed.). New York: John Wiley and Sons. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-00710-2</bdi>.</cite></li>

<li><cite id="CITEREFBillingsley2012" class="citation book cs1">Billingsley, Patrick (2012). Probability and Measure (Anniversary ed.). Hoboken, New Jersey: Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-118-12237-2</bdi>.</cite></li>

<li><cite id="CITEREFPark2018" class="citation book cs1">Park, Kun Il (2018). Fundamentals of Probability and Stochastic Processes with Applications to Communications. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-68074-3</bdi>.</cite></li>

<li><cite id="CITEREFRoss1993" class="citation book cs1"><a href="Sheldon_M._Ross" title="Sheldon M. Ross">Ross, Sheldon M.</a> (1993). Introduction to Probability Models (5th ed.). San Diego: Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-598455-3</bdi>.</cite></li></ul>

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