EX 3.24 check out DVD

• A library has an upper limit of 6 on the number of DVDs that can be checked out to an individual at one time.
• Consider only those who currently have DVDs checked out
• Let $X$ denote the number of DVDs checked out to a randomly selected individual.
• The pmf of $X$ is as follows:

$X$	1	2	3	4	5	6
$p(X)$	.30	.25	.15	.05	.10	.15

The expected value of $X$ is easily seen to be $\mu =2.85.$ The variance of $X$ is then

$$\begin{array}{rl}V(X)& ={\sigma }^2=\sum _{x=1}^6{\left(x-2.85\right)}^2\cdot p(x)\\ & ={\left(1-2.85\right)}^2(0.30)+{\left(2-2.85\right)}^2(0.25)+\cdots +{\left(6-2.85\right)}^2(0.15)\\ & =3.2275\end{array}$$

The standard deviation of $X$ is $\sigma =\sqrt{3.2275}$