An appliance dealer sells three different models of upright

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 the following pmf.

(a) Compute E(X), E(X²), and V(X). (Round your answers to two decimal places.)

(b) If the price of a freezer having capacity X cubic feet is 29X 8.5, what is the expected price paid by the next customer to buy a freezer? (Round your answer to the nearest whole number.)
dollars

(c) What is the variance of the price 29X 8.5 paid by the next customer? (Round your answer to the nearest whole number.)
dollars²

(d) Suppose that although the rated capacity of a freezer is X, the actual capacity is h(X) = X 0.02X². What is the expected actual capacity of the freezer purchased by the next customer? (Round your answer to three decimal places.)
ft³

Solution

(a)

E(X)=13.5*0.2+15.9*0.54+19.1*0.26 =16.25 ft³

E(X2)=13.5²*0.2+15.9²*0.54+19.1²*0.26=267.82 ft⁶

V(X)=E(X2)-E²(X)=3.69 ft⁶

(b)Expected price= E(29*X-8.5)=29*E(X)-8.5=29*16.25-8.5=463

(c)Variance of the price=V(29*X-8.5)=29²*V(X)=29*29*3.69=3103

(d)h(X)=X-0.02X²

E(h(X))=E(X)-0.02E(X²)=16.25-0.02*267.82=10.894 ft³