# Bernoulli Probability Distribution Family

## 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 $\displaystyle{ p }$ and value 0 with failure probability $\displaystyle{ q=1-p }$ . 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 $\displaystyle{ X }$ is a random variable with this distribution, we have: : $\displaystyle{ Pr(X=1) = 1 - Pr(X=0) = 1 - q = p.\! }$ A classic example of a Bernoulli experiment is a single toss of a coin. The coin might come up heads with probability $\displaystyle{ p }$ and tails with probability $\displaystyle{ 1-p }$ . The experiment is called fair if $\displaystyle{ p=0.5 }$ , indicating the origin of the terminology in betting (the bet is fair if both possible outcomes have the same probability).

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

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 than any other probability distribution, namely −2.

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

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