Variance of Binomial Distribution/Proof 3

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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

From Bernoulli Process as Binomial Distribution, we see that $X$ as defined here is the sum of the discrete random variables that model the Bernoulli distribution.

Each of the Bernoulli trials is independent of each other.

Hence we can use Sums of Variances of Independent Trials.

The Variance of Bernoulli Distribution is $p \left({1-p}\right)$ so the variance of $B \left({n, p}\right)$ is $n p\left({1-p}\right)$.

$\blacksquare$