Properties of the binomial distribution

We now find formulae for the mean and variance of the binomial 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

\[ E[X] = n\pi \quad\quad \text{and} \quad\quad \Var(X) = n\pi(1-\pi)\]

We first define a Bernoulli random variable for each of the different Bernoulli trials,

\[ B_i = \begin {cases} 1 & \quad \text{if the }i\text{'th trial is a success}\\[0.5em] 0 & \quad \text{if the }i\text{'th trial is a failure} \end {cases} \]

These \(n\) variables are independent since they are defined from independent Bernoulli trials and, from the properties of the Bernoulli distribution,

\[ E[B_i] = \pi \spaced{and} \Var(B_i) = \pi(1-\pi) \]

The total number of successes is the sum of these Bernoulli variables, \(X = \sum_{i=1}^n B_i\). Since this is the sum of the values in a random sample from a Bernoulli distribution, we can use the earlier result about the sum of values in a random sample,

\[ E[X] \;=\; E\left[\sum_{i=1}^n B_i\right] \;=\; n \times E[B_i] \;=\; n\pi \] \[ \Var(X) \;=\; \Var\left(\sum_{i=1}^n B_i\right) \;=\; n \times \Var[B_i] \;=\; n\pi(1-\pi) \]

These formulae give the mean and variance of the number of successes in a sequence of Bernoulli trials. We are often more interested in the proportion of successes,

\[ P = \frac X n \]

Since \(P\) is simply \(X\) multiplied by the constant \(\frac 1 n\), its mean and variance are closely related to those of \(X\).

Proportion of successes

The proportion of successes in a binomial experiment, \(P\), has mean and variance

\[ E[P] = \pi \spaced{and} \Var(P) = \frac {\pi(1-\pi)} n \]

These formulae can be easily obtained from the earlier general results about the mean and variance of a linear transformation of a random variable,

\[ E[a + b \times X] = a + b \times E[X] \] \[ \Var(a + b \times X) = b^2 \times \Var(X) \]

with \(a = 0\) and \(b = \frac 1 n\).