Variance of Binomial Distribution

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:
 * $$\operatorname{var} \left({X}\right) = n p \left({1-p}\right)$$

Proof 1
From the definition of Variance as Expectation of Square minus Square of Expectation:
 * $$\operatorname{var} \left({X}\right) = E \left({X^2}\right) - \left({E \left({X}\right)}\right)^2$$

From Expectation of Function of Discrete Random Variable:
 * $$E \left({X^2}\right) = \sum_{x \in \operatorname{Im} \left({X}\right)} x^2 \Pr \left({X = x}\right)$$

To simplify the algebra a bit, let $$q = 1 - p$$, so $$p+q = 1$$.

So:

$$ $$ $$ $$ $$ $$ $$ $$ $$ $$

Then:

$$ $$ $$

as required.

Proof 2
From Variance of Discrete Random Variable from P.G.F., we have:
 * $$\operatorname{var} \left({X}\right) = \Pi''_X \left({1}\right) + \mu - \mu^2$$

where $$\mu = E \left({x}\right)$$ is the expectation of $$X$$.

From the Probability Generating Function of Binomial Distribution, we have:
 * $$\Pi_X \left({s}\right) = \left({q + ps}\right)^n$$

where $$q = 1 - p$$.

From Expectation of Binomial Distribution, we have:
 * $$\mu = n p$$

Also from Expectation of Binomial Distribution, we have:
 * $$\Pi'_X \left({s}\right) = n p \left({q + ps}\right)^{n-1}$$

and so $$\Pi''_X \left({s}\right) = n \left({n-1}\right) p^2 \left({q + ps}\right)^{n-2}$$ from Power Rule for Derivatives and Chain Rule.

Putting $$s = 1$$ using the formula $$\Pi''_X \left({1}\right) + \mu - \mu^2$$:
 * $$\operatorname{var} \left({X}\right) = n \left({n-1}\right) p^2 + np - n^2p^2$$

and hence the result.