Number of successes
Since the number of successes, \(X\), can be written as the sum of \(n\) independent Bernoulli variables, we can find the binomial mean and variance from those of the Bernoulli distribution.
Binomial mean and variance
If \(X\) has a binomial distribution with probability function
\[ p(x)= {n \choose x} \pi^x(1-\pi)^{n-x} \quad \quad \text{for } x=0, 1, \dots, n \]then its mean and variance are
\[ E[X] = n\pi \quad\quad \text{and} \quad\quad \Var(X) = n\pi(1-\pi)\](Proved in full version)
Proportion of successes
The proportion of successes has a closely related distribution since
\[ P = \frac X n \]Proportion of successes
The proportion of successes in a binomial experiment, \(P\), has mean and variance
\[ E[P] = \pi \spaced{and} \Var(P) = \frac {\pi(1-\pi)} n \](Proved in full version)