A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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Chapter 5, Problem 5.17TE
If X has hazard rate
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Q1. A group of five applicants for a pair of identical jobs consists of three men and two
women. The employer is to select two of the five applicants for the jobs. Let S
denote the set of all possible outcomes for the employer's selection. Let A denote
the subset of outcomes corresponding to the selection of two men and B the subset
corresponding to the selection of at least one woman. List the outcomes in A, B,
AUB, AN B, and An B. (Denote the different men and women by M₁, M2, M3
and W₁, W2, respectively.)
Q3 (8 points)
Q3. A survey classified a large number of adults according to whether they were diag-
nosed as needing eyeglasses to correct their reading vision and whether they use
eyeglasses when reading. The proportions falling into the four resulting categories
are given in the following table:
Use Eyeglasses for Reading
Needs glasses Yes
No
Yes
0.44
0.14
No
0.02
0.40
If a single adult is selected from the large group, find the probabilities of the events
defined below. The adult
(a) needs glasses.
(b) needs glasses but does not use them.
(c) uses glasses whether the glasses are needed or not.
4. (i) Let a discrete sample space be given by
N = {W1, W2, W3, W4},
and let a probability measure P on be given by
P(w1) = 0.2, P(w2) = 0.2, P(w3) = 0.5, P(wa) = 0.1.
Consider the random variables X1, X2 → R defined by
X₁(w1) = 1, X₁(w2) = 2,
X2(w1) = 2, X2 (w2) = 2,
Find the joint distribution of X1, X2.
(ii)
X1(W3) = 1, X₁(w4) = 1,
X2(W3) = 1, X2(w4) = 2.
[4 Marks]
Let Y, Z be random variables on a probability space (, F, P).
Let the random vector (Y, Z) take on values in the set [0, 1] x [0,2] and let the
joint distribution of Y, Z on [0, 1] x [0,2] be given by
1
dPy,z (y, z) ==(y²z+yz2) dy dz.
harks 12 Find the distribution Py of the random variable Y.
[8 Marks]
Chapter 5 Solutions
A First Course in Probability (10th Edition)
Ch. 5 - Let X be a random variable with probability...Ch. 5 - Prob. 5.2PCh. 5 - Prob. 5.3PCh. 5 - The probability density function of X. the...Ch. 5 - Prob. 5.5PCh. 5 - Compute E[X] if X has a density function given by...Ch. 5 - The density function of X is given by...Ch. 5 - The lifetime in hours of an electronic tube is a...Ch. 5 - Consider Example 4b &I of Chapter 4 &I, but now...Ch. 5 - Trains headed for destination A arrive at the...
Ch. 5 - A point is chosen at random on a line segment of...Ch. 5 - A bus travels between the two cities A and B....Ch. 5 - You arrive at a bus stop at 10A.M., knowing that...Ch. 5 - Let X be a uniform (0, 1) random variable. Compute...Ch. 5 - If X is a normal random variable with parameters...Ch. 5 - The annual rainfall (in inches) in a certain...Ch. 5 - The salaries of physicians in a certain speciality...Ch. 5 - Suppose that X is a normal random variable with...Ch. 5 - Let be a normal random variable with mean 12 and...Ch. 5 - If 65 percent of the population of a large...Ch. 5 - Suppose that the height, in inches, of a...Ch. 5 - Every day Jo practices her tennis serve by...Ch. 5 - One thousand independent rolls of a fair die will...Ch. 5 - The lifetimes of interactive computer chips...Ch. 5 - Each item produced by a certain manufacturer is,...Ch. 5 - Two types of coins are produced at a factory: a...Ch. 5 - In 10,000 independent tosses of a coin, the coin...Ch. 5 - Twelve percent of the population is left handed....Ch. 5 - A model for the movement of a stock supposes that...Ch. 5 - An image is partitioned into two regions, one...Ch. 5 - a. A fire station is to be located along a road of...Ch. 5 - The time (in hours) required to repair a machine...Ch. 5 - If U is uniformly distributed on (0,1), find the...Ch. 5 - Jones figures that the total number of thousands...Ch. 5 - Prob. 5.35PCh. 5 - The lung cancer hazard rate (t) of a t-year-old...Ch. 5 - Suppose that the life distribution of an item has...Ch. 5 - If X is uniformly distributed over (1,1), find (a)...Ch. 5 - Prob. 5.39PCh. 5 - If X is an exponential random variable with...Ch. 5 - If X is uniformly distributed over(a,b), find a...Ch. 5 - Prob. 5.42PCh. 5 - Find the distribution of R=Asin, where A is a...Ch. 5 - Let Y be a log normal random variable (see Example...Ch. 5 - The speed of a molecule in a uniform gas at...Ch. 5 - Show that E[Y]=0P{Yy}dy0P{Yy}dy Hint: Show that...Ch. 5 - Show that if X has density function f. then...Ch. 5 - Prob. 5.4TECh. 5 - Use the result that for a nonnegative random...Ch. 5 - Prob. 5.6TECh. 5 - The standard deviation of X. denoted SD(X), is...Ch. 5 - Let X be a random variable that takes on values...Ch. 5 - Show that Z is a standard normal random variable;...Ch. 5 - Let f(x) denote the probability density function...Ch. 5 - Let Z be a standard normal random variable Z and...Ch. 5 - Use the identity of Theoretical Exercises 5.5 .Ch. 5 - The median of a continuous random variable having...Ch. 5 - The mode of a continuous random variable having...Ch. 5 - If X is an exponential random variable with...Ch. 5 - Compute the hazard rate function of X when X is...Ch. 5 - If X has hazard rate function X(t), compute the...Ch. 5 - Prob. 5.18TECh. 5 - If X is an exponential random variable with mean...Ch. 5 - Prob. 5.20TECh. 5 - Prob. 5.21TECh. 5 - Compute the hazard rate function of a gamma random...Ch. 5 - Compute the hazard rate function of a Weibull...Ch. 5 - Prob. 5.24TECh. 5 - Let Y=(Xv) Show that if X is a Weibull random...Ch. 5 - Let F be a continuous distribution function. If U...Ch. 5 - If X is uniformly distributed over (a,b), what...Ch. 5 - Consider the beta distribution with parameters...Ch. 5 - Prob. 5.29TECh. 5 - Prob. 5.30TECh. 5 - Prob. 5.31TECh. 5 - Let X and Y be independent random variables that...Ch. 5 - Prob. 5.33TECh. 5 - The number of minutes of playing time of a certain...Ch. 5 - For some constant c. the random variable X has the...Ch. 5 - Prob. 5.3STPECh. 5 - Prob. 5.4STPECh. 5 - The random variable X is said to be a discrete...Ch. 5 - Prob. 5.6STPECh. 5 - To be a winner in a certain game, you must be...Ch. 5 - A randomly chosen IQ test taker obtains a score...Ch. 5 - Suppose that the travel time from your home to...Ch. 5 - The life of a certain type of automobile tire is...Ch. 5 - The annual rainfall in Cleveland, Ohio, is...Ch. 5 - Prob. 5.12STPECh. 5 - Prob. 5.13STPECh. 5 - Prob. 5.14STPECh. 5 - The number of years that a washing machine...Ch. 5 - Prob. 5.16STPECh. 5 - Prob. 5.17STPECh. 5 - There are two types of batteries in a bin. When in...Ch. 5 - Prob. 5.19STPECh. 5 - For any real number y define byy+=y,ify00,ify0 Let...Ch. 5 - With (x) being the probability that a normal...Ch. 5 - Prob. 5.22STPECh. 5 - Letf(x)={13ex1313e(x1)ifx0if0x1ifx1 a. Show that f...Ch. 5 - Prob. 5.24STPE
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