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Bernoulli distribution

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In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,cite-ref-1[1] is the discrete probability distribution of a random variable which takes the value 1 with probability p {\displaystyle p} and the value 0 with probability q = 1 − − p {\displaystyle q=1-p} . Less formally, it can be thought of as a model for the set of possible outcomes of any single experiment that asks a yes–no question. Such questions lead to outcomes that are Boolean-valued: a single bit whose value is success/yes/true/one with probability p and failure/no/false/zero with probability q. It can be used to represent a (possibly biased) coin toss where 1 and 0 would represent "heads" and "tails", respectively, and p would be the probability of the coin landing on heads (or vice versa where 1 would represent tails and p would be the probability of tails). In particular, unfair coins would have p ≠ ≠ 1 / 2. {\displaystyle p\neq 1/2.}

The Bernoulli distribution is a special case of the binomial distribution where a single trial is conducted (so n would be 1 for such a binomial distribution). It is also a special case of the two-point distribution, for which the possible outcomes need not be 0 and 1.cite-ref-2[2]

Contents

• Properties

• Mean

• Variance

• Skewness

• Higher moments and cumulants

• Entropy and Fisher's Information

• Entropy

• Fisher's Information

• Related distributions

• See also

• References

• Further reading

• External links

Properties

If X {\displaystyle X} is a random variable with a Bernoulli distribution, then:

Pr ( X = 1 ) = p , Pr ( X = 0 ) = q = 1 − − p . {\displaystyle \Pr(X=1)=p,\Pr(X=0)=q=1-p.}

The probability mass function f {\displaystyle f} of this distribution, over possible outcomes k, is

f ( k ; p ) = { p if k = 1 , q = 1 − − p if k = 0. {\displaystyle f(k;p)={\begin{cases}p&{\text{if }}k=1,\\q=1-p&{\text{if }}k=0.\end{cases}}} cite-ref-0-3-0[3]

This can also be expressed as

f ( k ; p ) = p k ( 1 − − p ) 1 − − k for k ∈ ∈ { 0 , 1 } {\displaystyle f(k;p)=p^{k}(1-p)^{1-k}\quad {\text{for }}k\in \{0,1\}}

or as

f ( k ; p ) = p k + ( 1 − − p ) ( 1 − − k ) for k ∈ ∈ { 0 , 1 } . {\displaystyle f(k;p)=pk+(1-p)(1-k)\quad {\text{for }}k\in \{0,1\}.}

The Bernoulli distribution is a special case of the binomial distribution with n = 1. {\displaystyle n=1.} cite-ref-mccullagh1989ch422-4-0[4]

The kurtosis goes to infinity for high and low values of p , {\displaystyle p,} but for p = 1 / 2 {\displaystyle p=1/2} the two-point distributions including the Bernoulli distribution have a lower excess kurtosis, namely −2, than any other probability distribution.

The Bernoulli distributions for 0 ≤ ≤ p ≤ ≤ 1 {\displaystyle 0\leq p\leq 1} form an exponential family.

The maximum likelihood estimator of p {\displaystyle p} based on a random sample is the sample mean.

Mean

The expected value of a Bernoulli random variable X {\displaystyle X} is

E ⁡ ⁡ [ X ] = p {\displaystyle \operatorname {E} [X]=p}

This is because for a Bernoulli distributed random variable X {\displaystyle X} with Pr ( X = 1 ) = p {\displaystyle \Pr(X=1)=p} and Pr ( X = 0 ) = q {\displaystyle \Pr(X=0)=q} we find

E ⁡ ⁡ [ X ] = Pr ( X = 1 ) ⋅ ⋅ 1 + Pr ( X = 0 ) ⋅ ⋅ 0 = p ⋅ ⋅ 1 + q ⋅ ⋅ 0 = p . {\displaystyle \operatorname {E} [X]=\Pr(X=1)\cdot 1+\Pr(X=0)\cdot 0=p\cdot 1+q\cdot 0=p.} cite-ref-0-3-1[3]

Variance

The variance of a Bernoulli distributed X {\displaystyle X} is

Var ⁡ ⁡ [ X ] = p q = p ( 1 − − p ) {\displaystyle \operatorname {Var} [X]=pq=p(1-p)}

We first find

E ⁡ ⁡ [ X 2 ] = Pr ( X = 1 ) ⋅ ⋅ 1 2 + Pr ( X = 0 ) ⋅ ⋅ 0 2 {\displaystyle \operatorname {E} [X^{2}]=\Pr(X=1)\cdot 1^{2}+\Pr(X=0)\cdot 0^{2}}

= p ⋅ ⋅ 1 2 + q ⋅ ⋅ 0 2 = p = E ⁡ ⁡ [ X ] {\displaystyle =p\cdot 1^{2}+q\cdot 0^{2}=p=\operatorname {E} [X]}

From this follows

Var ⁡ ⁡ [ X ] = E ⁡ ⁡ [ X 2 ] − − E ⁡ ⁡ [ X ] 2 = E ⁡ ⁡ [ X ] − − E ⁡ ⁡ [ X ] 2 {\displaystyle \operatorname {Var} [X]=\operatorname {E} [X^{2}]-\operatorname {E} [X]^{2}=\operatorname {E} [X]-\operatorname {E} [X]^{2}}

= p − − p 2 = p ( 1 − − p ) = p q {\displaystyle =p-p^{2}=p(1-p)=pq} cite-ref-0-3-2[3]

With this result it is easy to prove that, for any Bernoulli distribution, its variance will have a value inside [ 0 , 1 / 4 ] {\displaystyle [0,1/4]} .

Skewness

The skewness is q − − p p q = 1 − − 2 p p q {\displaystyle {\frac {q-p}{\sqrt {pq}}}={\frac {1-2p}{\sqrt {pq}}}} . When we take the standardized Bernoulli distributed random variable X − − E ⁡ ⁡ [ X ] Var ⁡ ⁡ [ X ] {\displaystyle {\frac {X-\operatorname {E} [X]}{\sqrt {\operatorname {Var} [X]}}}} we find that this random variable attains q p q {\displaystyle {\frac {q}{\sqrt {pq}}}} with probability p {\displaystyle p} and attains − − p p q {\displaystyle -{\frac {p}{\sqrt {pq}}}} with probability q {\displaystyle q} . Thus we get

γ γ 1 = E ⁡ ⁡ [ ( X − − E ⁡ ⁡ [ X ] Var ⁡ ⁡ [ X ] ) 3 ] = p ⋅ ⋅ ( q p q ) 3 + q ⋅ ⋅ ( − − p p q ) 3 = 1 p q 3 ( p q 3 − − q p 3 ) = p q p q 3 ( q 2 − − p 2 ) = ( 1 − − p ) 2 − − p 2 p q = 1 − − 2 p p q = q − − p p q . {\displaystyle {\begin{aligned}\gamma _{1}&=\operatorname {E} \left[\left({\frac {X-\operatorname {E} [X]}{\sqrt {\operatorname {Var} [X]}}}\right)^{3}\right]\\&=p\cdot \left({\frac {q}{\sqrt {pq}}}\right)^{3}+q\cdot \left(-{\frac {p}{\sqrt {pq}}}\right)^{3}\\&={\frac {1}{{\sqrt {pq}}^{3}}}\left(pq^{3}-qp^{3}\right)\\&={\frac {pq}{{\sqrt {pq}}^{3}}}(q^{2}-p^{2})\\&={\frac {(1-p)^{2}-p^{2}}{\sqrt {pq}}}\\&={\frac {1-2p}{\sqrt {pq}}}={\frac {q-p}{\sqrt {pq}}}.\end{aligned}}}

Higher moments and cumulants

The raw moments are all equal because 1 k = 1 {\displaystyle 1^{k}=1} and 0 k = 0 {\displaystyle 0^{k}=0} .

E ⁡ ⁡ [ X k ] = Pr ( X = 1 ) ⋅ ⋅ 1 k + Pr ( X = 0 ) ⋅ ⋅ 0 k = p ⋅ ⋅ 1 + q ⋅ ⋅ 0 = p = E ⁡ ⁡ [ X ] . {\displaystyle \operatorname {E} [X^{k}]=\Pr(X=1)\cdot 1^{k}+\Pr(X=0)\cdot 0^{k}=p\cdot 1+q\cdot 0=p=\operatorname {E} [X].}

The central moment of order k {\displaystyle k} is given by

μ μ k = ( 1 − − p ) ( − − p ) k + p ( 1 − − p ) k . {\displaystyle \mu _{k}=(1-p)(-p)^{k}+p(1-p)^{k}.}

The first six central moments are

μ μ 1 = 0 , μ μ 2 = p ( 1 − − p ) , μ μ 3 = p ( 1 − − p ) ( 1 − − 2 p ) , μ μ 4 = p ( 1 − − p ) ( 1 − − 3 p ( 1 − − p ) ) , μ μ 5 = p ( 1 − − p ) ( 1 − − 2 p ) ( 1 − − 2 p ( 1 − − p ) ) , μ μ 6 = p ( 1 − − p ) ( 1 − − 5 p ( 1 − − p ) ( 1 − − p ( 1 − − p ) ) ) . {\displaystyle {\begin{aligned}\mu _{1}&=0,\\\mu _{2}&=p(1-p),\\\mu _{3}&=p(1-p)(1-2p),\\\mu _{4}&=p(1-p)(1-3p(1-p)),\\\mu _{5}&=p(1-p)(1-2p)(1-2p(1-p)),\\\mu _{6}&=p(1-p)(1-5p(1-p)(1-p(1-p))).\end{aligned}}}

The higher central moments can be expressed more compactly in terms of μ μ 2 {\displaystyle \mu _{2}} and μ μ 3 {\displaystyle \mu _{3}}

μ μ 4 = μ μ 2 ( 1 − − 3 μ μ 2 ) , μ μ 5 = μ μ 3 ( 1 − − 2 μ μ 2 ) , μ μ 6 = μ μ 2 ( 1 − − 5 μ μ 2 ( 1 − − μ μ 2 ) ) . {\displaystyle {\begin{aligned}\mu _{4}&=\mu _{2}(1-3\mu _{2}),\\\mu _{5}&=\mu _{3}(1-2\mu _{2}),\\\mu _{6}&=\mu _{2}(1-5\mu _{2}(1-\mu _{2})).\end{aligned}}}

The first six cumulants are

κ κ 1 = p , κ κ 2 = μ μ 2 , κ κ 3 = μ μ 3 , κ κ 4 = μ μ 2 ( 1 − − 6 μ μ 2 ) , κ κ 5 = μ μ 3 ( 1 − − 12 μ μ 2 ) , κ κ 6 = μ μ 2 ( 1 − − 30 μ μ 2 ( 1 − − 4 μ μ 2 ) ) . {\displaystyle {\begin{aligned}\kappa _{1}&=p,\\\kappa _{2}&=\mu _{2},\\\kappa _{3}&=\mu _{3},\\\kappa _{4}&=\mu _{2}(1-6\mu _{2}),\\\kappa _{5}&=\mu _{3}(1-12\mu _{2}),\\\kappa _{6}&=\mu _{2}(1-30\mu _{2}(1-4\mu _{2})).\end{aligned}}}

Entropy and Fisher's Information

Entropy

Entropy is a measure of uncertainty or randomness in a probability distribution. For a Bernoulli random variable X {\displaystyle X} with success probability p {\displaystyle p} and failure probability q = 1 − − p {\displaystyle q=1-p} , the entropy H ( X ) {\displaystyle H(X)} is defined as:

H ( X ) = E p ln ⁡ ⁡ ( 1 P ( X ) ) = − − [ P ( X = 0 ) ln ⁡ ⁡ P ( X = 0 ) + P ( X = 1 ) ln ⁡ ⁡ P ( X = 1 ) ] H ( X ) = − − ( q ln ⁡ ⁡ q + p ln ⁡ ⁡ p ) , q = P ( X = 0 ) , p = P ( X = 1 ) {\displaystyle {\begin{aligned}H(X)&=\mathbb {E} _{p}\ln({\frac {1}{P(X)}})=-[P(X=0)\ln P(X=0)+P(X=1)\ln P(X=1)]\\H(X)&=-(q\ln q+p\ln p),\quad q=P(X=0),p=P(X=1)\end{aligned}}}

The entropy is maximized when p = 0.5 {\displaystyle p=0.5} , indicating the highest level of uncertainty when both outcomes are equally likely. The entropy is zero when p = 0 {\displaystyle p=0} or p = 1 {\displaystyle p=1} , where one outcome is certain.

Fisher's Information

Fisher information measures the amount of information that an observable random variable X {\displaystyle X} carries about an unknown parameter p {\displaystyle p} upon which the probability of X {\displaystyle X} depends. For the Bernoulli distribution, the Fisher information with respect to the parameter p {\displaystyle p} is given by:

I ( p ) = 1 p q {\displaystyle {\begin{aligned}I(p)={\frac {1}{pq}}\end{aligned}}}

Proof:

• The Likelihood Function for a Bernoulli random variable X {\displaystyle X} is:

L ( p ; X ) = p X ( 1 − − p ) 1 − − X {\displaystyle {\begin{aligned}L(p;X)=p^{X}(1-p)^{1-X}\end{aligned}}}

This represents the probability of observing X {\displaystyle X} given the parameter p {\displaystyle p} .

• The Log-Likelihood Function is:

ln ⁡ ⁡ L ( p ; X ) = X ln ⁡ ⁡ p + ( 1 − − X ) ln ⁡ ⁡ ( 1 − − p ) {\displaystyle {\begin{aligned}\ln L(p;X)=X\ln p+(1-X)\ln(1-p)\end{aligned}}}

• The Score Function (the first derivative of the log-likelihood w.r.t. p {\displaystyle p} is:

∂ ∂ ∂ ∂ p ln ⁡ ⁡ L ( p ; X ) = X p − − 1 − − X 1 − − p {\displaystyle {\begin{aligned}{\frac {\partial }{\partial p}}\ln L(p;X)={\frac {X}{p}}-{\frac {1-X}{1-p}}\end{aligned}}}

• The second derivative of the log-likelihood function is:

∂ ∂ 2 ∂ ∂ p 2 ln ⁡ ⁡ L ( p ; X ) = − − X p 2 − − 1 − − X ( 1 − − p ) 2 {\displaystyle {\begin{aligned}{\frac {\partial ^{2}}{\partial p^{2}}}\ln L(p;X)=-{\frac {X}{p^{2}}}-{\frac {1-X}{(1-p)^{2}}}\end{aligned}}}

• Fisher information is calculated as the negative expected value of the second derivative of the log-likelihood:

I ( p ) = − − E [ ∂ ∂ 2 ∂ ∂ p 2 ln ⁡ ⁡ L ( p ; X ) ] = − − ( − − p p 2 − − 1 − − p ( 1 − − p ) 2 ) = 1 p ( 1 − − p ) = 1 p q {\displaystyle {\begin{aligned}I(p)=-E\left[{\frac {\partial ^{2}}{\partial p^{2}}}\ln L(p;X)\right]=-\left(-{\frac {p}{p^{2}}}-{\frac {1-p}{(1-p)^{2}}}\right)={\frac {1}{p(1-p)}}={\frac {1}{pq}}\end{aligned}}}

It is maximized when p = 0.5 {\displaystyle p=0.5} , reflecting maximum uncertainty and thus maximum information about the parameter p {\displaystyle p} .

Related distributions

• If X 1 , … … , X n {\displaystyle X_{1},\dots ,X_{n}} are independent, identically distributed (i.i.d.) random variables, all Bernoulli trials with success probability p, then their sum is distributed according to a binomial distribution with parameters n and p: ∑ ∑ k = 1 n X k ∼ ∼ B ⁡ ⁡ ( n , p ) {\displaystyle \sum _{k=1}^{n}X_{k}\sim \operatorname {B} (n,p)} (binomial distribution).cite-ref-0-3-3[3]

The Bernoulli distribution is simply B ⁡ ⁡ ( 1 , p ) {\displaystyle \operatorname {B} (1,p)} , also written as B e r n o u l l i ( p ) . {\textstyle \mathrm {Bernoulli} (p).}

• The categorical distribution is the generalization of the Bernoulli distribution for variables with any constant number of discrete values.

• The Beta distribution is the conjugate prior of the Bernoulli distribution.cite-ref-5[5]

• The geometric distribution models the number of independent and identical Bernoulli trials needed to get one success.

• If Y ∼ ∼ B e r n o u l l i ( 1 2 ) {\textstyle Y\sim \mathrm {Bernoulli} \left({\frac {1}{2}}\right)} , then 2 Y − − 1 {\textstyle 2Y-1} has a Rademacher distribution.