Index: The Book of Statistical Proofs ▷ Probability 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:

\[\label{eq:bin-var-s3} \mathrm{Var}(X) = \sum_{i=1}^{n} p \, (1-p) = n p \, (1-p) \; .\]
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Metadata: ID: P302 | shortcut: bin-var | author: JoramSoch | date: 2022-01-20, 15:19.