Mean and Variance of the Geometric Distribution

Compute the mean, variance, and standard deviation of a geometric random variable using $E[X]=\dfrac{1}{p}$ and $\operatorname{Var}(X)=\dfrac{1-p}{p^2}$ , both in direct computations and in applied word problems.

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Tutorial

Mean of the Geometric Distribution

Suppose $X \sim \operatorname{Geometric}(p)$ , where $X$ counts the number of independent trials needed to obtain the first success and each trial succeeds with probability $p \in (0,1]$ .

The mean (expected value) of $X$ is

E[X] = \dfrac{1}{p}.

Intuitively, if each trial succeeds with probability $p = \dfrac{1}{5}$ , we expect to wait about $5$ trials before the first success.

For example, if $p = 0.4$ , then

E[X] = \dfrac{1}{0.4} = 2.5.