Statistics for Business: Decision Making and Analysis, 3rd Edition

11.3 Properties of Binomial Random Variables

Mean and Variance

Because a binomial random variable is a sum of iid random variables, we can use the rules for expected values and variances from Chapter 10 to find its mean and variance. To find the mean of Y, the Addition Rule for Expected Values (the average of a sum is the sum of the averages) tells us that

$$\begin{array}{cc}\text{E}\left(Y\right)& =\text{E}\left(B_1\right)+\text{E}\left(B_2\right)+\cdots +\text{E}\left(B_n\right)\hfill \\ =p+p+\cdots +p\hfill \\ =np\hfill \end{array}$$

The mean of a binomial random variable is the number of trials times the probability of success. We expect a fraction p of the trials to result in a success.

For the variance, use the Addition Rule for Variances. Because Bernoulli random variables are independent of one another, the variance of a binomial random variable is the sum of the variances.

$$\begin{array}{c}\text{Var}\end{array}$$