Identify the expression for calculating the mean of a binomial distribution.

A random variable following a binomial distribution with success probability across trials has PMF



where .

The mean of the distribution is given by the expected value which is defined by



where the summation is carried out over the support of . So the mean is



Because this is a proper distribution, you have



which is a fact that will be used to evaluate the sum above.







Letting , this becomes



Observe that the remaining sum corresponds to the PMF of a new random variable which also follows a binomial distribution with success probability , but this time across trials. Therefore the sum evaluates to 1, and you're left with as the expression for the mean for .