Variance of Binomial Distribution/Proof 2

Theorem

Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$.

Then the variance of $X$ is given by:

$\var X = n p \paren {1 - p}$

Proof

From Variance of Discrete Random Variable from PGF:

$\var X = \map {\Pi_X} 1 + \mu - \mu^2$

where $\mu = \expect X$ is the expectation of $X$.

From the Probability Generating Function of Binomial Distribution:

$\map {\Pi_X} s = \paren {q + p s}^n$

where $q = 1 - p$.

From Expectation of Binomial Distribution:

$\mu = n p$

From Derivatives of PGF of Binomial Distribution:

$\map {\Pi_X} s = n \paren {n - 1} p^2 \paren {q + p s}^{n - 2}$

Setting $s = 1$ and using the formula $\map {\Pi_X} 1 + \mu - \mu^2$:

$\var X = n \paren {n - 1} p^2 + n p - n^2 p^2$

Hence the result.

$\blacksquare$