# Due Date: Friday, May 09, 2003, turn in after lecture

Correct solutions to any four problems carry full credit, but you must complete problem HW3_5. See the HW submission rules. On the front page include the following header.

• (HW_3_1)  [Sec. 3.1,  #4]  Let X = the number of nonzero digits in a randomly selected zip code. What are the possible values of X? Give three possible outcomes and their associated X values.

• (HW_3_2)  [Sec. 3.2,  #12]  Let X = the number of tires on a randomly selected automobile that are underinflated.
(a) Which of the following three p(x) functions is a legitimate probability mass function for X, and what are the other two not allowed?
 x 0 1 2 3 4 p1(x) 0.3 0.2 0.1 0.05 0.05 p2(x) 0.4 0.1 0.1 0.1 0.3 p3(x) 0.4 0.1 0.2 0.1 0.3

(b) For the legitimate pmf of part (a), compute P(2 ≤ X ≤ 4), P(X ≤ 2), and P(X ≠ 0).
(c) If p(x) = c (5 − x) for x = 0, 1, 2, 3, 4, what is the correct value of c?

• (HW_3_3) [Sec. 3.2,  #13]  A mail-order computer business has six telephone lines. Let X denote the number of lines in use at a specified time. Suppose the pmf of X is as given in the accompanying table. Calculate the probability of each of the following probabilities:  x 0 1 2 3 4 5 6 p(x) 0.1 0.15 0.2 0.25 0.2 0.06 0.04

(a) {at most 3 lines are in use}
(b) {fewer than 3 lines are in use}
(c) {at least 3 lines are in use}
(d) {between 2 and 5 lines, inclusive, are in use}
(e) {between 2 and 4 lines, inclusive, are not in use}
(f) {at least 4 lines are not in use}

• (HW_3_4) [Sec. 3.3,  #28] The pmf for X = the number of major defects on a randomly selected applicance of a certain type is given in the table below. Compute the following:
 x 0 1 2 3 4 p(x) 0.08 0.15 0.45 0.27 0.05
(a) E(X), the expected value of the R.V. X.
(b) V(X), the variance of X.
(c) SD(X), the standard deviation of X.
(d) V(X), using the shortcut formula, V(X)  =  E(X2 ) − µ2. Should equal the answer in part (b).

• (HW_3_5) [Sec. 3.3,  #31] An appliance dealer sells three different models of upright freezers having 13.5 ft3, 15.9 ft3, and 19.1 ft3 (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.
 x 13.5 15.9 19.1 p(x) 0.2 0.5 0.3

(a) Compute E(X), E(X2), E(X3), and V(X).
(b) If the price of a freezer having capacity X ft3 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? Interpret this value!
(d) Suppose that although the rated capacity of a freezer is X, the actual capacity is h(X) = X − 0.01X2, slightly under the advertised space. What is the expected actual capacity of the freezer purchased by the next customer?

Last modified on by .

Ivo D. Dinov, Ph.D., Departments of Statistics and Neurology, UCLA School of Medicine