An Appliance Dealer Sells Three Different Models of Upright Freezers $$\begin{array} { l c c c } \hline x & 13.5 & 15.9 & 19.1 \\\hline P ( x ) & .2 & .4 & .4 \\\hline\end{array}$$

Essay

An appliance dealer sells three different models of upright freezers having 13.5, 15.9, and 19.1 cubic feet of storage space, respectively. Let X = the amount of storage space purchased by the next customer to buy a freezer. Suppose that X has pmf $$\begin{array} { l c c c } \hline x & 13.5 & 15.9 & 19.1 \\\hline P ( x ) & .2 & .4 & .4 \\\hline\end{array}$$
a. Compute $E ( X ) , E \left( x ^ { 2 } \right) , \text { and } V ( X )$
b. If the price of a freezer having capacity X cubic feet is 25X - 8.5, what is the expected price paid by the next customer to buy a freezer?
c. What is the variance of the price 25X - 8.5 paid by the next customer?
d. Suppose that although the rated capacity of a freezer is X, the actual capacity is $h ( X ) = X - .01 X ^ { 2 }$
What is the expected actual capacity of the freezer purchased by the next customer?