Mean and Variance of the Binomial Distribution

Use the closed-form formulas $E[X]=np$ and $\mathrm{Var}(X)=np(1-p)$ to compute the mean, variance, and standard deviation of a binomial random variable -- both directly and from a modeling context, and to recover $n$ and $p$ from given moments.

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Mean and Variance of a Binomial Random Variable

If $X \sim \mathrm{Bin}(n, p)$ counts the number of successes in $n$ independent trials with success probability $p$ , then its mean and variance have simple closed forms.

The mean of $X$ is

E[X] = np.

The variance of $X$ is

\mathrm{Var}(X) = np(1-p).

The standard deviation is the square root of the variance:

\mathrm{SD}(X) = \sqrt{np(1-p)}.

For example, if $X \sim \mathrm{Bin}(10, 0.5)$ , then

\begin{align*} E[X] &= 10 \cdot 0.5 = 5, \\ \mathrm{Var}(X) &= 10 \cdot 0.5 \cdot 0.5 = 2.5, \\ \mathrm{SD}(X) &= \sqrt{2.5} \approx 1.58. \end{align*}

These formulas let us avoid summing the probability mass function term by term.