Variance of Binomial Distribution/Proof 2
Jump to navigation
Jump to search
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.
$\blacksquare$