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.

x 13.5 15.9 19.1
p(x) 0.17 0.57 0.26

a. Compute E(X), E(X2), and V(X).
b. If the price of a freezer having capacity X cubic feet is 28X − 8.5, what is the expected price paid by the next customer to buy a freezer? (Round your answer to the nearest whole number.)
c. What is the variance of the price 28X − 8.5 paid by the next customer? (Round your answer to the nearest whole number.)
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.)

a)

b)

And replacing the result from part a we got:

c)

d)

Step-by-step explanation:

For this case we have the following probability function given:

x 13.5 15.9 19.1

p(x) 0.17 0.57 0.26

The expected value of a random variable X is the n-th moment about zero of a probability density function f(x) if X is continuous, or the weighted average for a discrete probability distribution, if X is discrete.

The variance of a random variable X represent the spread of the possible values of the variable. The variance of X is written as Var(X).  

Part a

We can calculate the expected value with the following formula:

And if we replace we got:

For the second moment we can use this definition:

And if we replace we got:

The variance is defined:

Part b

For this case we define this new random variable Y = 28 X -8.5. And we want to find the expected value, so we have this:

And replacing the result from part a we got:

Part c

For the variance we can use the following property:

And using this formula we have:

The variance for a constant is 0 so then Var(8.5)=0 and Cov(28X, -8.5) = 0 since by properties if X is a random variable and a represent a constant , so then we just have this:

Using the following property we have:

Part d

For this case we define

And if we find the expected value we have this: