Bernoulli Probability Distribution Family

A Bernoulli Probability Distribution Family is binomial probability distribution family where n=1.

• • …
• Counter-Example(s):
  • Beta Distribution, Beta-Binomial Distribution.
• See: Bernoulli Trial, Logistic Regression.

---

---

References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Bernoulli_distribution Retrieved:2015-6-21.
  • In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jacob Bernoulli, is the probability distribution of a random variable which takes value 1 with success probability [math]\displaystyle{ p }[/math] and value 0 with failure probability [math]\displaystyle{ q=1-p }[/math] . It can be used, for example, to represent the toss of a (not necessarily fair) coin, where "1" is defined to mean "heads" and "0" is defined to mean "tails" (or vice versa).

    The Bernoulli distribution is a special case of the two-point distribution, for which the two possible outcomes need not be 0 and 1.

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Bernoulli_distribution#Properties Retrieved:2015-6-21.
  • If [math]\displaystyle{ X }[/math] is a random variable with this distribution, we have: : [math]\displaystyle{ Pr(X=1) = 1 - Pr(X=0) = 1 - q = p.\! }[/math] A classic example of a Bernoulli experiment is a single toss of a coin. The coin might come up heads with probability [math]\displaystyle{ p }[/math] and tails with probability [math]\displaystyle{ 1-p }[/math] . The experiment is called fair if [math]\displaystyle{ p=0.5 }[/math] , indicating the origin of the terminology in betting (the bet is fair if both possible outcomes have the same probability).

    The probability mass function [math]\displaystyle{ f }[/math] of this distribution, over possible outcomes k, is : [math]\displaystyle{ f(k;p) = \begin{cases} p & \text{if }k=1, \\[6pt] 1-p & \text {if }k=0.\end{cases} }[/math] This can also be expressed as : [math]\displaystyle{ f(k;p) = p^k (1-p)^{1-k}\!\quad \text{for }k\in\{0,1\}. }[/math] The expected value of a Bernoulli random variable [math]\displaystyle{ X }[/math] is : [math]\displaystyle{ E\left(X\right)=p }[/math] and its variance is : [math]\displaystyle{ \textrm{Var}\left(X\right)=p\left(1-p\right). }[/math] The Bernoulli distribution is a special case of the binomial distribution with [math]\displaystyle{ n = 1 }[/math] .

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

    The Bernoulli distributions for [math]\displaystyle{ 0 \le p \le 1 }[/math] form an exponential family.

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

---