1. A mail-order computer business has six telephone lines. Let X ...

. Using just the cdf, compute P (3 < X < 6) and P (4 < X).
Ans
P (3 < X < 6) = P [X = 3, 4, 6] = 0.1 + 0.05 + 0.25 = 0.3
P (4 < X) = P [X = 6, 12] = 0.55.
3. 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 probability mass function
x 13.5 15.9 19.1
p(x) .2 .5 .3
a. Compute E(X), E(X 2 ), and V (X).
Ans
E(X) = 16.38, E(X 2 ) = 272.298, and V (X) = 3.9936
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?
Ans
E[25X − 8.5] = (25)E[X] − 8.5 = 401,
c. What is the variance of the price 25X − 8.5 paid by the next
customer?
Ans
V [25X − 8.5] = [(25) 2 ]V [X] = 2496
d. Suppose that while the rated capacity of a freezer is X, the actual
capacity is h(X) = X − .01X 2 . What is the expected actual
capacity of the freezer purchased by the next customer?
Ans
E[h(X)] = E[X − .01X 2 ] = E[X] − (.01)E[X 2 ] = 13.66.
4. A chemical supply company currently has in stock 100 Kg of a certain
chemical, which it sells to customers 5 Kg lots. Let X = the number
of lots ordered by a randomly chosen customer, and suppose that X
has pmf
x 1 2 3 4
p(x) .2 .4 .3 .1
Compute E(X) and V (X). Then compute the expected number of
Kg’s left after the next customer’s order is shipped, and the variance
of the number of Kg’s left.
Ans
2