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introduction to probability statistics

Introduction To Probability And Statistics 3rd Edition William Mendenhall - Solutions

4. The distribution function of the random variable X is given(a) Plot this distribution function.(b) What is P{X > 12 }?(c) What is P{2 (d) What is P{X (e) What is P{X = 1}? 0 x 2 2 F(x)= x < 0 0 x < 1 1 x < 2 11 2 x
5. Suppose you are given the distribution function F of a random variable X . Explain how you could determine P{X = 1}. (Hint: You will need to use the concept of a limit.)
6. The amount of time, in hours, that a computer functions before breaking down is a continuous random variable with probability density function given byWhat is the probability that a computer will function between 50 and 150 hours before breaking down? What is the probability that it will
7. The lifetime in hours of a certain kind of radio tube is a random variable having a probability density function given byWhat is the probability that exactly 2 of 5 such tubes in a radio set will have to be replaced within the first 150 hours of operation? Assume that the events Ei , i = 1, 2,
8. If the density function of X equalsfindc. What is P{X > 2}? f(x)= = Ice 0 -2x 0
9. A bin of 5 transistors is known to contain 3 that are defective. The transistors are to be tested, one at a time, until the defective ones are identified. Denote by N1 the number of tests made until the first defective is spotted and by N2 the number of additional tests until the second
10. The joint probability density function of X and Y is given by(a) Verify that this is indeed a joint density function.(b) Compute the density function of X .(c) Find P{X > Y }.
11. Let X1, X2, . . . , Xn be independent random variables, each having a uniform distribution over (0, 1). LetM =maximum (X1, X2, . . . , Xn). Show that the distribution function of M, FM(·), is given byWhat is the probability density function of M? FM(x)=x", 0 x 1
12. The joint density of X and Y is given by(a) Compute the density of X .(b) Compute the density of Y .(c) Are X and Y independent? f(x,y) = 0 x > 0, y > 0 otherwise
13. The joint density of X and Y is(a) Compute the density of X .(b) Compute the density of Y .(c) Are X and Y independent? f(x, y) = 2 0
14. If the joint density function of X and Y factors into one part depending only on x and one depending only on y, show that X and Y are independent. That is, ifshow that X and Y are independent. f(x,y) = k(x)l(y), -
15. Is Problem 14 consistent with the results of Problems 12 and 13?
16. Suppose that X and Y are independent continuous random variables. Show thatwhere fY is the density function of Y , and FX is the distribution function of X . (a) P{X+Ya}= (b) P(X Y} = 00 Fx (a-y)fy (y) dy - Fx (y)fy (y) dy 00
17. When a current I (measured in amperes) flows through a resistance R (measured in ohms), the power generated (measured in watts) is given byW = I 2R. Suppose that I and R are independent random variables with densitiesDetermine the density function of W. 15x50 (x-1)x9= (x) f(x)=2x 0x1
18. In Example 4.3b, determine the conditional probability mass function of the size of a randomly chosen family containing 2 girls.
19. Compute the conditional density function of X given Y = y in (a) Problem 10 and (b) Problem 13.
20. Show that X and Y are independent if and only if(a) PX /Y(x/y) = pX (x) in the discrete case(b) fX /Y(x/y) = fX (x) in the continuous case
21. Compute the expected value of the random variable in Problem 1.
22. Compute the expected value of the random variable in Problem 3.
23. Each night different meteorologists give us the “probability” that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability p, then he or she will receive a score ofWe will then keep
24. An insurance company writes a policy to the effect that an amount of money A must be paid if some event E occurs within a year. If the company estimates that E will occur within a year with probability p, what should it charge the customer so that its expected profit will be 10 percent of A?
25. A total of 4 buses carrying 148 students from the same school arrive at a football stadium. The buses carry, respectively, 40, 33, 25, and 50 students. One of the students is randomly selected. Let X denote the number of students that were on the bus carrying this randomly selected student. One
26. Suppose that two teams play a series of games that end when one of them has won i games. Suppose that each game played is, independently, won by team A with probability p. Find the expected number of games that are played when i = 2.Also show that this number is maximized when p = 1 2 .
27. The density function of X is given byIf E[X] = 35 , finda, b. Ja + bx 0 x 1 f(x) = 0 otherwise
28. The lifetime in hours of electronic tubes is a random variable having a probability density function given byCompute the expected lifetime of such a tube. f(x) = axex, x0
29. Let X1, X2, . . . , Xn be independent random variables having the common density functionFind (a) E[Max(Xi , . . . , Xn)] and (b) E[Min(X1, . . . , Xn)]. f(x)= 1 0 < x
30. Suppose that X has density functionCompute E[X n] (a) by computing the density of Xn and then using the definition of expectation and (b) by using Proposition 4.5.1. 1 0 < x
31. The time it takes to repair a personal computer is a random variable whose density, in hours, is given byThe cost of the repair depends on the time it takes and is equal to 40 + 30 √x when the time is x. Compute the expected cost to repair a personal computer. 0 < x
32. If E[X] = 2 and E[X 2] = 8, calculate (a) E[(2+4X )2)] and (b) E[X 2+(X +1)2].
33. Ten balls are randomly chosen from an urn containing 17 white and 23 black balls. Let X denote the number of white balls chosen. Compute E[X ](a) by defining appropriate indicator variables Xi , i = 1, . . . , 10 so that(b) by defining appropriate indicator variables Yi ,= 1, . . . , 17 so that
34. If X is a continuous random variable having distribution function F , then its median is defined as that value of m for which F (m) = 1/2 Find the median of the random variables with density function(a) f (x) = e−x , x ≥ 0;(b) f (x) = 1, 0 ≤ x ≤ 1.
35. The median, like the mean, is important in predicting the value of a random variable. Whereas it was shown in the text that the mean of a random variable is the best predictor from the point of view of minimizing the expected value of the square of the error, the median is the best predictor if
36. We say that mp is the 100p percentile of the distribution function F if F (mp) = p Find mp for the distribution having density function f(x)=2e2x, x0
37. A community consists of 100 married couples. If during a given year 50 of the members of the community die, what is the expected number of marriages that remain intact? Assume that the set of people who die is equally likely to be any of thegroups of size 50. (Hint: For i = 1, . . . , 100 let
38. Compute the expectation and variance of the number of successes in n independent trials, each of which results in a success with probability p. Is independence necessary?
39. Suppose that X is equally likely to take on any of the values 1, 2, 3, 4. Compute(a) E[X ] and (b) Var(X ).
40. Let pi = P{X = i} and suppose that p1 +p2 +p3 = 1. If E[X] = 2, what values of p1, p2, p3 (a) maximize and (b) minimize Var(X )?
41. Compute the mean and variance of the number of heads that appear in 3 flips of a fair coin.
42. Argue that for any random variable XWhen does one have equality? E[X] (E[X])
43. A random variable X , which represents the weight (in ounces) of an article, has density function given by f (z),(a) Calculate the mean and variance of the random variable X .(b) The manufacturer sells the article for a fixed price of $2.00. He guarantees to refund the purchase money to any
44. Suppose that the Rockwell hardness X and abrasion loss Y of a specimen (coded data) have a joint density given by(a) Find the marginal densities of X and Y .(b) Find E(X ) and Var(X ). fxy (u,v) = [u+v for 0 u, v 1 0 otherwise
45. A product is classified according to the number of defects it contains and the factory that produces it. Let X1 and X2 be the random variables that represent the number of defects per unit (taking on possible values of 0, 1, 2, or 3) and the factory number (taking on possible values 1 or 2),
46. A machine makes a product that is screened (inspected 100 percent) before being shipped. The measuring instrument is such that it is difficult to read between 1 and 113(coded data). After the screening process takes place, the measured dimension has density(a) Find the value of k.(b) What
47. Verify Equation 4.7.4.
48. Prove Equation 4.7.5 by using mathematical induction.
49. Let X have variance σ2 x and let Y have variance σ2 y . Starting withUsing the result that Var(Z) = 0 implies that Z is constant, argue that if Corr(X , Y ) = 1 or −1 then X and Y are related by Y = a + bx where the sign of b is positive when the correlation is 1 and negative when it is
50. Consider n independent trials, each of which results in any of the outcomes i, i =1, 2, 3, with respective probabilities p1, p2, p3,3i=1 pi = 1. Let Ni denote the number of trials that result in outcome i, and show that Cov(N1,N2) = −np1p2.Also explain why it is intuitive that this covariance
51. In Example 4.5f, compute Cov(Xi , Xj ) and use this result to show that Var(X ) = 1.
52. If X1 and X2 have the same probability distribution function, show thatNote that independence is not being assumed. Cov(X1 X2, X+X2) = 0
53. Suppose that X has density functionCompute the moment generating function of X and use your result to determine its mean and variance. Check your answer for the mean by a direct calculation. f(x)=ex, x>0
54. If the density function of X isdetermine E[etX ]. Differentiate to obtain E[X n] and then check your answer. f(x)=1, 0 < x < 1
55. Suppose that X is a random variable with mean and variance both equal to 20.What can be said about P{0 ≤ X ≤ 40}?
56. From past experience, a professor knows that the test score of a student taking her final examination is a random variable with mean 75.(a) Give an upper bound to the probability that a student’s test score will exceed 85.Suppose in addition the professor knows that the variance of a
57. Let X and Y have respective distribution functions FX and FY , and suppose that for some constants a and b > 0,(a) Determine E[X ] in terms of E[Y ].(b) Determine Var(X ) in terms of Var(Y ).Hint: X has the same distribution as what other random variable? x-a Fx(x) = Fy b
An election will be held next week and, by polling a sample of the voting population, we are trying to predict whether the Republican or Democratic candidate will prevail. Which of the following methods of selection is likely to yield a representative sample?(a) Poll all people of voting age
The approach used in Problem 1(e) led to a disastrous prediction in the 1936 presidential election, in which Franklin Roosevelt defeated Alfred Landon by a landslide. A Landon victory had been predicted by the Literary Digest. The magazine based its prediction on the preferences of a sample of
A researcher is trying to discover the average age at death for people in the United States today. To obtain data, the obituary columns of the New York Times are read for 30 days, and the ages at death of people in the United States are noted. Do you think this approach will lead to a
To determine the proportion of people in your town who are smokers, it has been decided to poll people at one of the following local spots:(a) the pool hall;(b) the bowling alley;(c) the shopping mall;(d) the library.Which of these potential polling places would most likely result in a reasonable
A university plans on conducting a survey of its recent graduates to determine information on their yearly salaries. It randomly selected 200 recent graduates and sent them questionnaires dealing with their present jobs. Of these 200, however, only 86 were returned. Suppose that the average of the
An article reported that a survey of clothing worn by pedestrians killed at night in traffic accidents revealed that about 80 percent of the victims were wearing darkcolored clothing and 20 percent were wearing light-colored clothing. The conclusion drawn in the article was that it is safer to wear
Critique Graunt’s method for estimating the population of London. What implicit assumption is he making?
The London bills of mortality listed 12,246 deaths in 1658. Supposing that a survey of London parishes showed that roughly 2 percent of the population died that year, use Graunt’s method to estimate London’s population in 1658.
Suppose you were a seller of annuities in 1662 when Graunt’s book was published.Explain how you would make use of his data on the ages at which people were dying.
Based on Graunt’s mortality table:(a) What proportion of people survived to age 6?(b) What proportion survived to age 46?(c) What proportion died between the ages of 6 and 36?
The wing stroke frequencies of two species of Euglossine bees were recorded for a sample of n = 4 Euglossa mandibularis Friese (species 1) and n = 6 Euglossa im- perialis Cockerell (species 2). The frequencies are listed in Table 15.3. Can you con- clude that the distributions of wing strokes
An experiment was conducted to compare the strengths of two types of kraft papers: one a standard kraft paper of a specified weight and the other the same standard kraft paper treated with a chemical substance. Ten pieces of each type of paper, randomly selected from production, produced the
15.1 Suppose you want to use the Wilcoxon rank sum test to detect a shift in distribution I to the right of dis- tribution 2 based on samples of size n = 6 and n = 8.a. Should you use T or T as the test statistic?b. What is the rejection region for the test if a = 0.05?c. What is the rejection
15.2 Refer to Exercise 15.1. Suppose the alternative hypothesis is that distribution 1 is shifted either to the left or to the right of distribution 2.a. Should you use T or T as the test statistic?b. What is the rejection region for the test if a = 0.05?c. What is the rejection region for the test
15.3 Observations from two random and independent samples, drawn from populations 1 and 2, are given here. Use the Wilcoxon rank sum test to determine whether population 1 is shifted to the left of population 2.a. State the null and alternative hypotheses to be tested.b. Rank the combined sample
15.4 Independent random samples of size n = 20 and n 25 are drawn from non-normal populations 1 and 2. The combined sample is ranked and T = 252. Use the large-sample approximation to the Wilcoxon rank sum test to determine whether there is a difference in the two population distributions.
15.5 Suppose you wish to detect a shift in distribution 1 to the right of distribution 2 based on sample sizes n = 12 and n = 14. If T = 193, what do you conclude? Use a = 0.05.
15.6 Alzheimer's Disease In some tests of healthy, elderly men, a new drug has restored their memory almost to that of young people. It will soon be tested on patients with Alzheimer's disease, the fatal brain disorder that destroys the mind. According to Dr. Gary Lynch, the drug, called ampakine
15.7 Alzheimer's, continued Refer to Exercise 15.6. Suppose that two more groups of 10 men each are tested on the number of nonsense syllables they can remember after 5 minutes. However, this time the 65-70-year-olds are given a mild dose of ampakine CX-516. Do the data provide sufficient evidence
15.8 Dissolved O Content The observations in the table are dissolved oxygen contents in water. The higher the dissolved oxygen content, the greater the ability of a river, lake, or stream to support aquatic life. In this experiment, a pollution-control inspector suspected that a river community was
15.9 Eye Movement In an investigation of the visual scanning behaviour of deaf children, measurements of eye movement were taken on nine deaf and nine hearing children. The table gives the eye-movement rates and their ranks (in parentheses). Does it appear that the distributions of eye-movement
15.10 Comparing NFL Quarterbacks How does Aaron Rodgers, quarterback for the 2011 Super Bowl winners, the Minnesota Vikings, compare to Drew Brees, quarterback for the 2010 Super Bowl winners, the New Orleans Saints? The table below shows the number of completed passes for each athlete during the
15.11 Weights of Turtles The weights of turtles caught in two different lakes were measured to compare the effects of the two lake environments on turtle growth. All the turtles were the same age and were tagged before being released into the lakes. The weights for n = 10 tagged turtles caught in
15.12 Chemotherapy Cancer treatment by means of chemicals-chemotherapy-kills both cancerous and normal cells. In some instances, the toxicity of the cancer drug-that is, its effect on normal cells can be reduced by the simultaneous injection of a second drug. A study was conducted to determine
The numbers of defective electrical fuses produced by two production lines, A and B, were recorded daily for a period of 10 days, with the results shown in Table 15.6. The response variable, the number of defective fuses, has an exact binomial distribution with a large number of fuses produced per
A production superintendent claims that there is no difference between the employee accident rates for the day versus the evening shifts in a large manufacturing plant. The number of accidents per day is recorded for both the day and evening shifts for n = 100 days. It is found that the number of
15.13 Suppose you wish to use the sign test to test Ha p>0.5 for a paired-difference experiment with n = 25 pairs.a. State the practical situation that dictates the alternative hypothesis given.b. Use Table 1 in Appendix I to find values of a (a
15.14 Repeat the instructions of Exercise 15.13 for Ha p 0.5.
15.15 Repeat the instructions of Exercises 15.13 and 15.14 for n = 10, 15, and 20.
15.16 A paired-difference experiment was conducted to compare two populations. The data are shown in the table. Use a sign test to deter- mine whether the population distributions are different.a. State the null and alternative hypotheses for the test.b. Determine an appropriate rejection region
15.17 Property Values In Exercise 10.47, you compared the property evaluations of two tax assessors, A and B. Their assessments for eight properties are shown in the table:a. Use the sign test to determine whether the data present sufficient evidence to indicate that one of the assessors tends to
15.18 Gourmet Cooking Two gourmets, A and B, rated 22 meals on a scale of 1 to 10. The data are shown in the table. Do the data provide sufficient evidence to indicate that one of the gourmets tends to give higher ratings than the other? Test by using the sign test with a value of a near 0.05.a.
15.19 Lead Levels in Blood A study reported in the American Journal of Public Health (Science News)- the first to follow blood lead levels in law-abiding handgun hobbyists using indoor firing ranges- documents a significant risk of lead poisoning.* Lead exposure measurements were made on 17 members
15.20 Recovery Rates Clinical data concerning the effectiveness of two drugs intreating a particular disease were collected from 10 hospitals. The numbers of patients treated with the drugs varied from one hospital to another. You want to know whether the data present sufficient evidence to
An experiment was conducted to compare the densities of cakes prepared from two dif- ferent cake mixes, A and B. Six cake pans received batter A, and six received batter B. Expecting a variation in oven temperature, the experimenter placed an A and a B cake side by side at six different locations
15.21 Suppose you wish to detect a difference in the locations of two population distributions based on a paired-difference experiment consisting of n = 30 pairs.a. Give the null and alternative hypotheses for the Wilcoxon signed-rank test.b. Give the test statistic.c. Give the rejection region for
15.22 Refer to Exercise 15.21. Suppose you wish to detect only a shift in distribution 1 to the right of distribution 2.a. Give the null and alternative hypotheses for the Wilcoxon signed-rank test.b. Give the test statistic.c. Give the rejection region for the test for a = 0.05. =d. If 7 249, what
15.23 Refer to Exercise 15.21. Conduct the test using the large-sample z test. Compare your results with the nonparametric test results in Exercise 15.22, part d.
15.24 Refer to Exercise 15.22. Conduct the test using the large-sample z test. Compare your results with the nonparametric test results in Exercise 15.21, part d.
15.25 Refer to Exercise 15.16 and data set EX1516. The data in this table are from a paired-difference experiment with n = 7 pairs of observations.a. Use Wilcoxon's signed-rank test to determine whether there is a significant difference between the two populations.b. Compare the results of part a
15.26 Property Values II In Exercise 15.17, you used the sign test to determine whether the data provided sufficient evidence to indicate a difference in the distributions of property assessments for assessors A and B.a. Use the Wilcoxon signed-rank test for a paired experiment to test the null
15.27 Machine Breakdowns The number of machine breakdowns per month was recorded for nine months on two identical machines, A and B, used to make wire rope:a. Do the data provide sufficient evidence to indicate a difference in the monthly breakdown rates for the two machines? Test by using a value
15.28 Gourmet Cooking II Refer to the comparison of gourmet meal ratings in Exercise 15.18, and use the Wilcoxon signed-rank test to determine whether the data provide sufficient evidence to indicate a difference in the ratings of the two gourmets. Test by using a value of a near 0.05. Compare the
15.29 Traffic Control Two methods for controlling traffic, A and B, were used at each of n = 12 intersections for a period of 1 week, and the numbers of accidents that occurred during this time period were recorded. The order of use (which method would be employed for the first week) was selected
15.30 Jigsaw Puzzles Eight people were asked to perform a simple puzzle-assembly task under normal conditions and under stressful conditions. During the stressful time, a mild shock was delivered to subjects 3 minutes after the start of the experiment and every 30 seconds thereafter until the task
15.31 Images and Word Recall A psychol- ogy class performed an experiment to determine whether a recall score in which instructions to form images of 25 words were given differs from an initial recall score for which no imagery instructions were given. Twenty students participated in the experiment

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