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Bernoulli distribution - Wikipedia

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Bernoulli distribution

Probability mass function

Funzione di densità di una variabile casuale normale

Three examples of Bernoulli distribution:

  {\displaystyle P(x=0)=0{.}2} and {\displaystyle P(x=1)=0{.}8}

  {\displaystyle P(x=0)=0{.}8} and {\displaystyle P(x=1)=0{.}2}

  {\displaystyle P(x=0)=0{.}5} and {\displaystyle P(x=1)=0{.}5}

Parameters

{\displaystyle 0\leq p\leq 1}

{\displaystyle q=1-p}
Support {\displaystyle k\in \{0,1\}}
PMF {\displaystyle {\begin{cases}q=1-p&{\text{if }}k=0\\p&{\text{if }}k=1\end{cases}}}
CDF {\displaystyle {\begin{cases}0&{\text{if }}k<0\\1-p&{\text{if }}0\leq k<1\\1&{\text{if }}k\geq 1\end{cases}}}
Mean {\displaystyle p}
Median {\displaystyle {\begin{cases}0&{\text{if }}p<1/2\\\left[0,1\right]&{\text{if }}p=1/2\\1&{\text{if }}p>1/2\end{cases}}}
Mode {\displaystyle {\begin{cases}0&{\text{if }}p<1/2\\0,1&{\text{if }}p=1/2\\1&{\text{if }}p>1/2\end{cases}}}
Variance {\displaystyle p(1-p)=pq}
MAD {\displaystyle 2p(1-p)=2pq}
Skewness {\displaystyle {\frac {q-p}{\sqrt {pq}}}}
Excess kurtosis {\displaystyle {\frac {1-6pq}{pq}}}
Entropy {\displaystyle -q\ln q-p\ln p}
MGF {\displaystyle q+pe^{t}}
CF {\displaystyle q+pe^{it}}
PGF {\displaystyle q+pz}
Fisher information {\displaystyle {\frac {1}{pq}}}

In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,[1] is the discrete probability distribution of a random variable which takes the value 1 with probability {\displaystyle p} and the value 0 with probability {\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 {\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.[2]

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

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

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

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

This can also be expressed as

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

or as

{\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 {\displaystyle n=1.}[4]

The kurtosis goes to infinity for high and low values of {\displaystyle p,} but for {\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 {\displaystyle 0\leq p\leq 1} form an exponential family.

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

The probability mass distribution function of a Bernoulli experiment along with its corresponding cumulative distribution function.

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

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

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

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

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

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

We first find

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

From this follows

{\displaystyle \operatorname {Var} [X]=\operatorname {E} [X^{2}]-\operatorname {E} [X]^{2}=\operatorname {E} [X]-\operatorname {E} [X]^{2}}
{\displaystyle =p-p^{2}=p(1-p)=pq}[3]

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

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

{\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

[edit]

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

{\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 {\displaystyle k} is given by

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

The first six central moments are

{\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 {\displaystyle \mu _{2}} and {\displaystyle \mu _{3}}

{\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

{\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

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

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

Fisher's Information

[edit]

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

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

Proof:

  • The Likelihood Function for a Bernoulli random variable{\displaystyle X} is:
{\displaystyle {\begin{aligned}L(p;X)=p^{X}(1-p)^{1-X}\end{aligned}}}

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

  • The Log-Likelihood Function is:
{\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. {\displaystyle p} is:
{\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:
{\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:
{\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 {\displaystyle p=0.5}, reflecting maximum uncertainty and thus maximum information about the parameter {\displaystyle p}.

The Bernoulli distribution is simply {\displaystyle \operatorname {B} (1,p)}, also written as {\textstyle \mathrm {Bernoulli} (p).}
  1. ^ Uspensky, James Victor (1937). Introduction to Mathematical Probability. New York: McGraw-Hill. p. 45. OCLC 996937.
  2. ^ Dekking, Frederik; Kraaikamp, Cornelis; Lopuhaä, Hendrik; Meester, Ludolf (9 October 2010). A Modern Introduction to Probability and Statistics (1 ed.). Springer London. pp. 43–48. ISBN 9781849969529.
  3. ^ a b c d Bertsekas, Dimitri P. (2002). Introduction to Probability. Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν. Belmont, Mass.: Athena Scientific. ISBN 188652940X. OCLC 51441829.
  4. ^ McCullagh, Peter; Nelder, John (1989). Generalized Linear Models, Second Edition. Boca Raton: Chapman and Hall/CRC. Section 4.2.2. ISBN 0-412-31760-5.
  5. ^ Orloff, Jeremy; Bloom, Jonathan. "Conjugate priors: Beta and normal" (PDF). math.mit.edu. Retrieved October 20, 2023.
  • Johnson, N. L.; Kotz, S.; Kemp, A. (1993). Univariate Discrete Distributions (2nd ed.). Wiley. ISBN 0-471-54897-9.
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