Bernoulli Distribution

The Bernoulli distribution models the probability distribution of a random variable that takes the value \(1\) with probability \(p\) (success) and the value \(0\) with probability \(1-p\) (failure) in a single independent Bernoulli trial.

A Bernoulli random variable is denoted as the following:

\[ X \sim \text{Bernoulli}(p) \]

Probability Mass Function (PMF)

The pmf of a Bernoulli random variable can be denoted as the following:

\[\begin{split} p_X(x) = \begin{cases} p, & \text{if } x = 1 \\ 1-p, & \text{if } x = 0 \\ 0, & \text{otherwise} \end{cases} =\begin{cases} p^x(1-p)^{1-x}, & \text{if } x = 0, 1 \\ 0, & \text{otherwise} \end{cases} \end{split}\]

Expectation

The expectation for a Bernoulli random variable can be derived as follows:

\[\begin{split} \begin{align} EX &= \sum_{x=0}^1 x \ p_X(x) \\ &= 0p^0(1-p)^1 + 1p^1(1-p)^0 \\ &= p \end{align} \end{split}\]

Variance

The variance for a Bernoulli random variable can be calculated:

\[ Var(X) = EX^2 - (EX)^2 \]

We must first find \(EX^2\):

\[\begin{split} \begin{align} EX^2 &= \sum_{x=0}^1 x^2 p_X(x) \\ &= 0^2p^0(1-p)^1 + 1^2 p^1(1-p)^0 \\ &= p \end{align} \end{split}\]

With this, we can find \(Var(X)\):

\[\begin{split} \begin{align} Var(X) &= EX^2 - (EX)^2 \\ &= p - p^2 = p(1-p) \end{align} \end{split}\]