3.4.3 The Mean and Variance of X

For $n=1$ , the binomial distribution becomes the Bernoulli distribution. From EX 3.18 expected value of Bernoulli random variable, the mean value of a Bernoulli variable is $\mu =p$, so the expected number of $S$ ‘s on any single trial is $p$. Since a binomial experiment consists of $n$ trials, intuition suggests that for

• $X\sim \mathrm{Bin}\left(n,p\right)$
• $E\left(X\right)=np$ the product of the number of trials and the probability of success on a single trial. The expression for $V\left(X\right)$ is not so intuitive.

Proposition

If $X\sim \mathrm{Bin}\left(n,p\right)$ , then

• $E\left(X\right)=np$,
• $V\left(X\right)=np\left(1-p\right)=npq$
• ${\sigma }_X=\sqrt{npq}$
  • where $q=1-p$

Thus, calculating the mean and variance of a binomial rv does not necessitate evaluating summations. The proof of the result for $E\left(X\right)$ is sketched in Exercise 64.

EX 3.33 purchase with credit card