Mean and Variance of the Bernoulli Distribution

Derive and apply the formulas for the mean, variance, and standard deviation of a Bernoulli random variable, and recover the parameter $p$ from a given variance.

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Tutorial

Mean of a Bernoulli Random Variable

A Bernoulli random variable $X$ with parameter $p$ takes the value $1$ with probability $p$ and the value $0$ with probability $q = 1-p.$

By the definition of expected value for a discrete random variable,

E[X] = \sum\limits_x x \cdot P(X=x) = 0 \cdot (1-p) + 1 \cdot p = p.

So the mean of a Bernoulli random variable with parameter $p$ is

\mu = E[X] = p.

For example, if $X \sim \text{Bernoulli}(0.25),$ then $E[X] = 0.25.$