STAT 110 A, Probability & Statistics for Engineers I UCLA Statistics, Spring 2003

http://www.stat.ucla.edu/~dinov/courses_students.html

HOMEWORK 3

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) 0.10 0.15 0.20 0.25 0.20 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?

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

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

x	0	1	2	3	4	5	6
p(x)	0.10	0.15	0.20	0.25	0.20	0.06	0.04

x	0	1	2	3	4
p(x)	0.08	0.15	0.45	0.27	0.05

x	13.5	15.9	19.1
p(x)	0.2	0.5	0.3