A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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Chapter 5, Problem 5.33P
If U is uniformly distributed on
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Exercise 1
Mateo is the star player of a certain soccer team and is getting ready for a very important match after
spending several months recovering from an injury. If the player is injured again during this match, the
probability that his team wins is 0.32. If Mateo is not injured, the probability that his team loses is 0.18. In
addition, according to the team doctor, the probability that the player gets injured during the match is 0.15.
a. Draw a probability tree that properly represents the situation described above. Clearly label the
probabilities on each branch of the tree.
b. What is the probability that Mateo's team wins the match?
c. If Mateo's team loses the match, what is the probability that Mateo was injured?
d. Consider the event "Mateo is injured" and the event "Mateo's team wins the match." Are these events
independent? Clearly justify your answer.
A company devoted to the production and distribution of craft beer has decided to run a
quality-control check on a batch of 330 mL Porter beer bottles. A random sample of the
contents of 52 bottles was taken; the volume (in millilitres, mL) was measured and is shown
below:
Dato
Volumen
(ml)
Volumen
Volumen
Volumen
Dato
Dato
Dato
(ml)
(ml)
(ml)
333
14
326
27
330
40
329
2
328
15
331
28
329
41
327
3
335
16
331
29
329
42
323
4
330
17
333
30
332
43
330
5
331
18
332
31
334
44
332
6
326
19
327
32
336
45
333
7
330
20
330
33
328
46
328
8
329
21
328
34
326
47
329
9
332
22
325
35
330
48
327
10
334
23
330
36
332
49
331
11
333
24
334
37
333
50
333
12
336
25
332
38
331
51
328
13
327
26
325
39
334
52
336
Exercise 4
A company that manufactures engine parts has developed a new type of piston made from aluminium-
silicon alloys and other materials. The firm wishes to examine the mechanical properties of these pistons.
Specifically, it is interested in evaluating the yield strength of a piston while it is subjected to a constant
temperature of 250 °C. Here, the yield strength is the maximum stress, measured in megapascals (MPa), that
the piston can withstand before it deforms permanently.
From the first production batches, 15 pistons were randomly selected. Each was tested to determine its yield
strength (in MPa), giving the following results:
83.2, 90.1, 86.7, 102.4, 95.9, 91.3, 88.1, 84.6, 93.2, 92.6, 100.4, 86.5, 89.2, 96.8, 91.7.
Assuming that the yield strength of this type of piston follows a Normal distribution, construct a 96 %
confidence interval for the variance, σ², of the pistons' yield strength.
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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