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introduction to probability statistics

Introduction To Probability And Statistics For Engineers And Scientists 3rd Edition Sheldon M. Ross - Solutions

27. Do the data in Problem 26 appear to be approximately normal? For parts (c) and(d) of this problem, compare the approximate proportions given by the empirical rule with the actual proportions.
28. Would you expect that a histogram of the weights of all the members of a health club would be approximately normal?
29. Use the data of Problem 16.(a) Compute the sample mean and sample median.(b) Are the data approximately normal?(c) Compute the sample standard deviation s.(d) What percentage of the data fall within ¯x ±1.5s?(e) Compare your answer in part (d) to that given by the empirical rule.(f) Compare
30. Use the data concerning the first 10 states listed in the table given in Problem 15.(a) Draw a scatter diagram relating the 1992 and 1993 salaries.(b) Determine the sample correlation coefficient.
31. The following table gives the median salaries for recent U.S. doctorate recipients, categorized by scientific field and type of employment. Draw a scatter diagram relating salaries in private firms to those in government, and determine the sample correlation coefficient.
32. Use the table to find the sample correlation coefficients between salaries in(a) government and universities(b) private firms and universities. Median salaries for recent U.S. doctorate recipients (1-3 years after degree), by sector of employment: 1999 (Dollars) Private Tenure-track at
33. Using data on the first 10 cities listed in Table 2.5, draw a scatter diagram and find the sample correlation coefficient between the January and July temperatures.
34. Verify property 3 of the sample correlation coefficient.
35. Verify property 4 of the sample correlation coefficient.
36. In a study of children in grades 2 through 4, a researcher gave each student a reading test. When looking at the resulting data the researcher noted a positive correlation between a student’s reading test score and height. The researcher concluded that taller children read better because they
A total of 28 percent of American males smoke cigarettes, 7 percent smoke cigars, and 5 percent smoke both cigars and cigarettes. What percentage of males smoke neither cigars nor cigarettes?
Two balls are “randomly drawn” from a bowl containing 6 white and 5 black balls. What is the probability that one of the drawn balls is white and the other black?
Mr. Jones has 10 books that he is going to put on his bookshelf. Of these, 4 are mathematics books, 3 are chemistry books, 2 are history books, and 1 is a language book. Jones wants to arrange his books so that all the books dealing with the same subject are together on the shelf. How many
A class in probability theory consists of 6 men and 4 women. An exam is given and the students are ranked according to their performance. Assuming that no two students obtain the same score, (a) how many different rankings are possible? (b) If all rankings are considered equally likely, what is the
A committee of size 5 is to be selected from a group of 6 men and 9 women.If the selection is made randomly, what is the probability that the committee consists of 3 men and 2 women?
From a set of n items a random sample of size k is to be selected. What is the probability a given item will be among the k selected?
A basketball team consists of 6 black and 6 white players. The players are to be paired in groups of two for the purpose of determining roommates. If the pairings are done at random, what is the probability that none of the black players will have a white roommate?
If n people are present in a room, what is the probability that no two of them celebrate their birthday on the same day of the year? How large need n be so that this probability is less than 1/2 ?
A bin contains 5 defective (that immediately fail when put in use), 10 partially defective (that fail after a couple of hours of use), and 25 acceptable transistors.A transistor is chosen at random from the bin and put into use. If it does not immediately fail, what is the probability it is
The organization that Jones works for is running a father–son dinner for those employees having at least one son. Each of these employees is invited to attend along with his youngest son. If Jones is known to have two children, what is the conditional probability that they are both boys given
Ms. Perez figures that there is a 30 percent chance that her company will set up a branch office in Phoenix. If it does, she is 60 percent certain that she will be made manager of this new operation. What is the probability that Perez will be a Phoenix branch office manager?
An insurance company believes that people can be divided into two classes — those that are accident prone and those that are not. Their statistics show that an accident-prone person will have an accident at some time within a fixed 1-year period with probability .4, whereas this probability
Reconsider Example 3.7a and suppose that a new policy holder has an accident within a year of purchasing his policy. What is the probability that he is accident prone?
In answering a question on a multiple-choice test, a student either knows the answer or she guesses. Let p be the probability that she knows the answer and 1−p the probability that she guesses. Assume that a student who guesses at the answer will be correct with probability 1/m, where m is the
A laboratory blood test is 99 percent effective in detecting a certain disease when it is, in fact, present. However, the test also yields a “false positive” result for 1 percent of the healthy persons tested. (That is, if a healthy person is tested, then, with probability .01, the test result
At a certain stage of a criminal investigation, the inspector in charge is 60 percent convinced of the guilt of a certain suspect. Suppose now that a new piece of evidence that shows that the criminal has a certain characteristic (such as left-handedness, baldness, brown hair, etc.) is uncovered.
Let us now suppose that the new evidence is subject to different possible interpretations, and in fact only shows that it is 90 percent likely that the criminal possesses this certain characteristic. In this case, how likely would it be that the suspect is guilty (assuming, as before, that he has
A plane is missing and it is presumed that it was equally likely to have gone down in any of three possible regions. Let 1−αi denote the probability the plane will be found upon a search of the ith region when the plane is, in fact, in that region, i =1, 2, 3. (The constants αi are called
A card is selected at random from an ordinary deck of 52 playing cards. If A is the event that the selected card is an ace and H is the event that it is a heart, then A and H are independent, since P(AH) = 1 52 , while P(A) = 4 52 and P(H) = 13 52 .
If we let E denote the event that the next president is a Republican and F the event that there will be a major earthquake within the next year, then most people would probably be willing to assume that E and F are independent. However, there would probably be some controversy over whether it is
Two fair dice are thrown. Let E7 denote the event that the sum of the dice is 7. Let F denote the event that the first die equals 4 and let T be the event that the second die equals 3. Now it can be shown (see Problem 36) that E7 is independent of F and that E7 is also independent of T; but clearly
A system composed of n separate components is said to be a parallel system if it functions when at least one of the components functions. (See Figure 3.7.) For such a system, if component i, independent of other components, functions with probability pi , i = 1, . . . , n, what is the probability
A set of k coupons, each of which is independently a type j coupon with probability is collected. Find the probability that the set contains a type j coupon given that it contains a type i, i = j. n Pj, j=1 Pj 1,
1. A box contains three marbles — one red, one green, and one blue. Consider an experiment that consists of taking one marble from the box, then replacing it in the box and drawing a second marble from the box. Describe the sample space.Repeat for the case in which the second marble is drawn
2. An experiment consists of tossing a coin three times. What is the sample space of this experiment? Which event corresponds to the experiment resulting in more heads than tails?
3. Let S = {1, 2, 3, 4, 5, 6, 7}, E = {1, 3, 5, 7}, F = {7, 4, 6}, G = {1, 4}. Find(a) EF; (c) EG c ; (e) Ec (F ∪ G );(b) E ∪ FG; (d) EF c ∪ G; (f ) EG ∪ FG.
4. Two dice are thrown. Let E be the event that the sum of the dice is odd, let F be the event that the first die lands on 1, and let G be the event that the sum is 5.Describe the events EF, E ∪ F, FG, EF c , EFG.
5. Asystem is composed of four components, each of which is either working or failed.Consider an experiment that consists of observing the status of each component, and let the outcome of the experiment be given by the vector (x1, x2, x3, x4) where xi is equal to 1 if component i is working and is
6. Let E, F, G be three events. Find expressions for the events that of E, F, G(a) only E occurs;(b) both E and G but not F occur;(c) at least one of the events occurs;(d) at least two of the events occur;(e) all three occur;(f ) none of the events occurs;(g) at most one of them occurs;(h) at most
7. Find simple expressions for the events(a) E ∪ Ec ;(b) EE c ;(c) (E ∪ F )(E ∪ F c );(d) (E ∪ F )(Ec ∪ F )E ∪ F c );(e) (E ∪ F )(F ∪ G).
8. Use Venn diagrams (or any other method) to show that(a) EF ⊂ E, E ⊂ E ∪ F ;(b) if E ⊂ F then F c ⊂ Ec ;(c) the commutative laws are valid;(d) the associative laws are valid;(e) F = FE ∪ FEc ;(f ) E ∪ F = E ∪ EcF ;(g) DeMorgan’s laws are valid.
9. For the following Venn diagram, describe in terms of E, F, and G the events denoted in the diagram by the Roman numerals I through VII. E II F S I III IV VII V VI G
10. Show that if E ⊂F then P(E ) ≤ P(F ). (Hint: Write F as the union of two mutually exclusive events, one of them being E.)
11. Prove Boole’s inequality, namely that " PEP(E) \=1 i=1
12. If P(E ) = .9 and P(F ) = .9, show that P(EF )≥.8. In general, prove Bonferroni’s inequality, namely that P(EF) P(E)+P(F)-1
13. Prove that(a) P(EF c ) = P(E )−P(EF )(b) P(EcF c ) = 1−P(E )−P(F )+P(EF )
14. Show that the probability that exactly one of the events E or F occurs is equal to P(E )+P(F )−2P(EF ).
15. Calculate93,96,7 2,7 5,10 7.
16. Show that Now present a combinatorial argument for the foregoing by explaining why a choice of r items from a set of size n is equivalent to a choice of n−r items from that set. (7)=(27)
17. Show thatFor a combinatorial argument, consider a set of n items and fix attention on one of these items. How many different sets of size r contain this item, and how many do not? (n-1 (n-1 (*)=(})+(7) r
18. A group of 5 boys and 10 girls is lined up in random order — that is, each of the 15! permutations is assumed to be equally likely.(a) What is the probability that the person in the 4th position is a boy?(b) What about the person in the 12th position?(c) What is the probability that a
19. Consider a set of 23 unrelated people. Because each pair of people shares the same birthday with probability 1/365, and there are23 2= 253 pairs, why isn’t the probability that at least two people have the same birthday equal to 253/365?
20. A town contains 4 television repairmen. If 4 sets break down, what is the probability that exactly 2 of the repairmen are called? What assumptions are you making?
21. A woman has n keys, of which one will open her door. If she tries the keys at random, discarding those that do not work, what is the probability that she will open the door on her kth try? What if she does not discard previously tried keys?
22. A closet contains 8 pairs of shoes. If 4 shoes are randomly selected, what is the probability that there will be (a) no complete pair and (b) exactly 1 complete pair?
23. Of three cards, one is painted red on both sides; one is painted black on both sides;and one is painted red on one side and black on the other. A card is randomly chosen and placed on a table. If the side facing up is red, what is the probability that the other side is also red?
24. A couple has 2 children. What is the probability that both are girls if the eldest is a girl?
27. There are two local factories that produce radios. Each radio produced at factory A is defective with probability .05, whereas each one produced at factory B is defective with probability .01. Suppose you purchase two radios that were produced at the same factory, which is equally likely to
28. A red die, a blue die, and a yellow die (all six-sided) are rolled. We are interested in the probability that the number appearing on the blue die is less than that appearing on the yellow die which is less than that appearing on the red die. (That is, if B (R) [Y ] is the number appearing on
29. You ask your neighbor to water a sickly plant while you are on vacation. Without water it will die with probability .8; with water it will die with probability .15. You are 90 percent certain that your neighbor will remember to water the plant.(a) What is the probability that the plant will be
30. Two balls, each equally likely to be colored either red or blue, are put in an urn.At each stage one of the balls is randomly chosen, its color is noted, and it is then returned to the urn. If the first two balls chosen are colored red, what is the probability that(a) both balls in the urn are
31. A total of 600 of the 1,000 people in a retirement community classify themselves as Republicans, while the others classify themselves as Democrats. In a local election in which everyone voted, 60 Republicans voted for the Democratic candidate, and 50 Democrats voted for the Republican
34. Prostate cancer is the most common type of cancer found in males. As an indicator of whether a male has prostate cancer, doctors often perform a test that measures the level of the PSA protein (prostate specific antigen) that is produced only by the prostate gland. Although higher PSA levels
35. Suppose that an insurance company classifies people into one of three classes —good risks, average risks, and bad risks. Their records indicate that the probabilities that good, average, and bad risk persons will be involved in an accident over a 1-year span are, respectively, .05, .15, and
36. A pair of fair dice is rolled. Let E denote the event that the sum of the dice is equal to 7.(a) Show that E is independent of the event that the first die lands on 4.(b) Show that E is independent of the event that the second die lands on 3.
37. The probability of the closing of the ith relay in the circuits shown is given by pi , i = 1, 2, 3, 4, 5. If all relays function independently, what is the probability that a current flows between A and B for the respective circuits?
38. An engineering system consisting of n components is said to be a k-out-ofn system (k ≤ n) if the system functions if and only if at least k of the ncomponents function. Suppose that all components function independently of each other.(a) If the ith component functions with probability Pi , i
39. Five independent flips of a fair coin are made. Find the probability that(a) the first three flips are the same;(b) either the first three flips are the same, or the last three flips are the same;(c) there are at least two heads among the first three flips, and at least two tails among the last
40. Suppose that n independent trials, each of which results in any of the outcomes 0, 1, or 2, with respective probabilities .3, .5, and .2, are performed. Find the probability that both outcome 1 and outcome 2 occur at least once. (Hint: Consider the complementary probability.)
41. Aparallel system functions whenever at least one of its components works. Consider a parallel system of n components, and suppose that each component independently works with probability 12. Find the conditional probability that component 1 works, given that the system is functioning.
42. A certain organism possesses a pair of each of 5 different genes (which we will designate by the first 5 letters of the English alphabet). Each gene appears in 2 forms (which we designate by lowercase and capital letters). The capital letter will be assumed to be the dominant gene in the sense
43. Three prisoners are informed by their jailer that one of them has been chosen at random to be executed, and the other two are to be freed. Prisoner A asks the jailer to tell him privately which of his fellow prisoners will be set free, claiming that there would be no harm in divulging this
44. Although both my parents have brown eyes, I have blue eyes. What is the probability that my sister has blue eyes?
45. A set of k coupons, each of which is independently a type j coupon with probability pj ,nj=1 pj = 1, is collected. Find the probability that the set contains either a type i or a type j coupon.
Letting X denote the random variable that is defined as the sum of two fair dice, thenIn other words, the random variable X can take on any integral value between 2 and 12 and the probability that it takes on each value is given by Equation 4.1.1. Since X must take on some value, we must havewhich
EXAMPLE 4.1b Suppose that an individual purchases two electronic components each of which may be either defective or acceptable. In addition, suppose that the four possible results — (d, d ), (d, a), (a, d ), (a,a) — have respective probabilities .09, .21, .21, .49[where (d, d ) means that both
Suppose the random variable X has distribution function What is the probability that X exceeds 1? 0 F(x) = x 0 1- exp(-x) x>0
Consider a random variable X that is equal to 1, 2, or 3. If we know thatthen it follows (since p(1) + p(2) + p(3) = 1) that A graph of p(x) is presented in Figure 4.1. ■The cumulative distribution function F can be expressed in terms of p(x) by p(1) = 2 and p(2) =
Suppose that X is a continuous random variable whose probability density function is given by(a) What is the value of C?(b) Find P{X > 1}. f(x): C(4x-2x) 0 < x
The joint density function of X and Y is given byCompute (a) P{X > 1, Y (b) P{X (c) P{X 2ee2 f(x,y) = 0 0
Suppose that X and Y are independent random variables having the common density functionFind the density function of the random variable X /Y . f(x)= otherwise
Suppose that the successive daily changes of the price of a given stock are assumed to be independent and identically distributed random variables with probability mass function given byThen the probability that the stock’s price will increase successively by 1, 2, and 0 points in the next three
If we know, in Example 4.3b, that the family chosen has one girl, compute the conditional probability mass function of the number of boys in the family.
Suppose that p(x, y), the joint probability mass function of X and Y , is given byCalculate the conditional probability mass function of X given that Y = 1. p(0, 0) = 4, p(0, 1) = .2, p(1, 0) = .1, p(1, 1) = .3.
The joint density of X and Y is given byCompute the conditional density of X , given that Y = y, where 0 f(x, y) = |x(2-x-y) 0 < x < 1,0 < y < 1 0 otherwise
Find E[X ] where X is the outcome when we roll a fair die.
If I is an indicator random variable for the event A, that is, ifthen E[I] = 1P(A) + 0P(Ac ) = P(A)Hence, the expectation of the indicator random variable for the event A is just the probability that A occurs. 1 = 1 if A occurs 0 if A does not occur
Suppose that you are expecting a message at some time past 5 P.M. From experience you know that X , the number of hours after 5 P.M. until the message arrives, is a random variable with the following probability density function:The expected amount of time past 5 P.M. until the message arrives is
Suppose X has the following probability mass functionCalculate E[X2]. p(0)=.2, p(1)=.5, p(2) = .3
The time, in hours, it takes to locate and repair an electrical breakdown in a certain factory is a random variable— call it X — whose density function is given byIf the cost involved in a breakdown of duration x is x3, what is the expected cost of such a breakdown? fx(x) = 1 if 0
Applying Proposition 4.5.1 to Example 4.5a yieldswhich, of course, checks with the result derived in Example 4.5a. E[X2] 02(0.2) (12) (0.5) + (22) (0.3) = 1.7 =
A construction firm has recently sent in bids for 3 jobs worth (in profits)10, 20, and 40 (thousand) dollars. If its probabilities of winning the jobs are respectively.2, .8, and .3, what is the firm’s expected total profit?
A secretary has typed N letters along with their respective envelopes. The envelopes get mixed up when they fall on the floor. If the letters are placed in the mixed-up envelopes in a completely random manner (that is, each letter is equally likely to end up in any of the envelopes), what is the
Suppose there are 20 different types of coupons and suppose that each time one obtains a coupon it is equally likely to be any one of the types. Compute the expected number of different types that are contained in a set for 10 coupons.
Compute Var(X ) when X represents the outcome when we roll a fair die.
Compute the variance of the sum obtained when 10 independent rolls of a fair die are made.
Compute the variance of the number of heads resulting from 10 independent tosses of a fair coin.
Suppose that it is known that the number of items produced in a factory during a week is a random variable with mean 50.(a) What can be said about the probability that this week’s production will exceed 75?(b) If the variance of a week’s production is known to equal 25, then what can be said
1. Five men and 5 women are ranked according to their scores on an examination.Assume that no two scores are alike and all 10! possible rankings are equally likely.Let X denote the highest ranking achieved by a woman (for instance, X = 2 if the top-ranked person was male and the next-ranked person
3. In Problem 2, if the coin is assumed fair, for n = 3, what are the probabilities associated with the values that X can take on?

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