Index: The Book of Statistical ProofsProbability Distributions ▷ Univariate discrete distributions ▷ Binomial distribution ▷ Variance

Theorem: Let $X$ be a random variable following a binomial distribution:

$\label{eq:bin} X \sim \mathrm{Bin}(n,p) \; .$

Then, the variance of $X$ is

$\label{eq:bin-var} \mathrm{Var}(X) = n p \, (1-p) \; .$

Proof: By definition, a binomial random variable is the sum of $n$ independent and identical Bernoulli trials with success probability $p$. Therefore, the variance is

$\label{eq:bin-var-s1} \mathrm{Var}(X) = \mathrm{Var}(X_1 + \ldots + X_n)$

and because variances add up under independence, this is equal to

$\label{eq:bin-var-s2} \mathrm{Var}(X) = \mathrm{Var}(X_1) + \ldots + \mathrm{Var}(X_n) = \sum_{i=1}^{n} \mathrm{Var}(X_i) \; .$

With the variance of the Bernoulli distribution, we have:

$\label{eq:bin-var-s3} \mathrm{Var}(X) = \sum_{i=1}^{n} p \, (1-p) = n p \, (1-p) \; .$
Sources:

Metadata: ID: P302 | shortcut: bin-var | author: JoramSoch | date: 2022-01-20, 15:19.