The University of Melbourne School of Mathematics and Statistics MAST20004 Probability STUDENT NAME: EMAIL ADDRESS: Cover image: Jakob Bernoulli (1654 – 1705, Basel) Jakob Bernoulli was a Swiss mathematician who made significant contributions to algebra, calculus, mechanics, calculus of variations, geometry, infinite series, and probability. His most renowned work was the derivation of the law of large numbers, which put simply, states that the average proportion of times an event occurs approaches the theoretical probability of the event, as the number of trials increases. This compilation has been made in accordance with the provisions of Part VB of the copyright act for the teaching purposes of the University. For use of students of the University of Melbourne enrolled in the second year subject MAST20004 Probability. Table of Contents
Problem Sheets
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1 Topic 1 – Axioms of Probability …………………………………………………………………. 1 Topic 2 – Conditional Probability and Independence ……………………………………… 3 Topic 3 – Random Variables and Distribution Functions ……………………………….. 7 Topic 4 – Special Probability Distributions …………………………………………………. 14 Topic 5 – Bivariate Random Variables ……………………………………………………….. 23 Topic 6 – Sums of Independent Random Variables and Limit Theorems ………… 33 Topic 7 – Stochastic Processes ………………………………………………………………….. 38 Answers to Problem Sheets ……………………………………………………………………………… 41 Topic 1 – Axioms of Probability …………………………………………………………………. 41 Topic 2 – Conditional Probability and Independence …………………………………….. 43 Topic 3 – Random Variables and Distribution Functions ………………………………. 46 Topic 4 – Special Probability Distributions …………………………………………………. 51 Topic 5 – Bivariate Random Variables ……………………………………………………….. 58 Topic 6 – Sums of Independent Random Variables and Limit Theorems ………….. 64 Topic 7 – Stochastic Processes ……………………………………………………………………. 67 MAST20004 Probability Semester 1, 2021 Problem Sheet 1 Axioms of Probability 1. Sample Space Define a sample space for the experiment of putting three different books on a shelf in random order. If two of these three books are a two-volume dictionary, describe the event that these volumes stand in increasing order side-by-side (i.e., volume I precedes volume II and next to each other). 2. Events Let E, F , and G be three events ; explain the meaning of the two relations E ∪F ∪G = G and E ∩ F ∩G = G. 3. More Events Prove that the event B is impossible if and only if for every event A, A = (B ∩ Ac) ∪ (Bc ∩ A). 4. Cards In an experiment, cards are drawn, one by one, at random and successively from an ordinary deck of 52 cards. Let An be the event that no face card or ace appears on the first n 1 draws, and the nth draw is an ace. In terms of Ans, find an expression for the event that an ace appears before a face card, if (a) the cards are drawn with replacement ; (b) they are drawn without replacement. 5. Event Identities Let A and B be two events. Prove the following relations by the elementwise method. (a) (A(A ∩B)) ∪B = A ∪B ; (b) (A ∪B)(A ∩B) = (A ∩Bc) ∪ (Ac ∩B). 6. Infinite Sequence of Sets Let {A1, A2, A3, . . .} be a sequence of events of a sample space S. Find a sequence {B1, B2, B3, . . .} of mutually exclusive events such that for all n ≥ 1, ∪ni=1Ai = ∪ni=1Bi. 1 7. Hiring A company has only one position with three highly qualified applicants : John, Barbara, and Marty. However, because the company has only a few women employees, Barbara’s chance to be hired is 20% higher than John’s and 20% higher than Marty’s. Find the probability that Barbara will be hired. 8. Probability Statements Which of the following statements is true If a statement is true, prove it. If it is false, give a counterexample. (a) If P(A) + P(B) + P(C) = 1, then A, B, and C are mutually exclusive events ; (b) If P(A ∪B ∪ C) = 1, then A, B, and C are mutually exclusive events. 9. Probability Identity Let A, B, and C be three events. Prove that P(A ∪B ∪ C) = P(A) + P(B) + P(C) P(A ∩B) P(A ∩ C) P(B ∩ C) + P(A ∩B ∩ C). 10. Random Numbers A number is selected randomly from the set {1, 2, . . . , 1000}. What is the probability that (a) it is divisible by 3 but not by 5 ; (b) it is divisible neither by 3 nor by 5 11. Voting For a Democratic candidate to win an election, she must win districts I, II, and III. Polls have shown that the probability of winning I and III is 0.55, losing II but not I is 0.34, and losing II and III but not I is 0.15. Find the probability that this candidate will win all three districts. (Draw a Venn diagram.) 12. Boole’s Inequality Let A1, A2, A3, . . . be a sequence of events of a sample space. Prove that P ( ∞ n=1 An ) ≤ ∞∑ n=1 P(An). This is called Boole’s inequality. 2 MAST20004 Probability Semester 1, 2021 Problem Sheet 2 Conditional Probability and Independence 13. Dice Suppose that two fair dice have been tossed and the total of their top faces is found to be divisible by 5. What is the probability that both of them have landed 5 14. Movies The cinemas of a town are showing seven comedies and nine dramas. Brian has seen five of the movies. If the first three movies he has seen are dramas, what is the probability that the last two are comedies Assume that Brian chooses the shows at random and sees each movie at most once. 15. Conditional Probability Properties Let Ω be the sample space of an experiment, and let B be an event of Ω with P(B) > 0. Prove that (a) P(A|B) ≥ 0 for any event A Ω ; (b) P(Ω|B) = 1 ; (c) If A1, A2, . . . is a sequence of mutually exclusive events, then P ( ∞ i=1 Ai |B ) = ∞∑ i=1 P(Ai|B). 16. Defective and Nondefective Items If eight defective and 12 nondefective items are inspected one by one, at random and without replacement, what is the probability that (a) the first four items inspected are defective ; (b) from the first three items at least two are defective 17. Credit Rating Suppose that 75% of all people with credit records improve their credit ratings within three years. Suppose that 18% of the population at large have poor credit records, and of those only 30% will improve their credit ratings within three years. What percentage of the people who 3 will improve their credit records within the next three years are the ones who currently have good credit ratings 18. Cards From an ordinary deck of 52 cards, cards are drawn one by one, at random and without replacement. What is the probability that the fourth heart is drawn on the tenth draw Hint : Let F denote the event that in the first nine draws there are exactly three hearts, and E be the event that the tenth draw is a heart. Use P(F ∩ E) = P(F )P(E|F ). 19. Proof (a) Show that if P(A) = 1, then P(A ∩B) = P(B) ; (b) Prove that any event A with P(A) = 0 or P(A) = 1 is independent of every event B. 20. More Proofs (a) Show that if an event A is independent of itself, then P(A) = 0 or 1 ; (b) Show that if A and B are independent and A B, then either P(A) = 0 or P(B) = 1. 21. Matching A fair die is rolled six times. If on the ith roll, 1 ≤ i ≤ 6, the outcome is i, we say that a match has occurred. What is the probability that at least one match occurs 22. Families From the set of all families with three children a family is selected at random. Let A be the event that “the family has children of both sexes” and B be the event that “there is at most one girl in the family”. Are A and B independent Answer the same question for families with two children and families with four children. Assume that for any family size all sex distributions have equal probabilities. 23. Independent Events Let {A1, A2, . . . , An} be an independent set of events and P(Ai) = pi, 1 ≤ i ≤ n. (a) What is the probability that at least one of the events A1, A2, . . . , An occurs (b) What is the probability that none of the events A1, A2, . . . , An occurs 24. Urns and Balls An urn contains two red and four white balls. Balls are drawn from the urn successively, at random and with replacement. What is the probability that exactly three whites occur in the first five trials 25. A Before B Let Ω be the sample space of a repeatable experiment. Let A and B be mutually exclusive events of Ω. Prove that, in independent trials of this experiment, the event A occurs before the event B with probability P(A)/(P(A) + P(B)). 4 26. Guns A person has six guns. The probability of hitting a target when these guns are properly aimed and fired is 0.6, 0.5, 0.7, 0.9, 0.7, and 0.8, respectively. What is the probability of hitting a target if a gun is selected at random, properly aimed, and fired 27. Strokes Of the patients in a hospital, 20% of those with, and 35% of those without myocardial infarction have had strokes. If 40% of the patients have had myocardial infarction, what percent of the patients have had strokes 28. Lost Spades (a) One of the cards of an ordinary deck of 52 cards is lost. What is the probability that a random card drawn from this deck is a spade (b) Two cards from an ordinary deck of 52 cards are missing. What is the probability that a random card drawn from this deck is a spade 29. Proof Let B be an event of a sample space Ω with P(B) > 0. For a subset A of Ω, define Q(A) = P(A|B). By Question 15 we know that Q is a probability function. For E and F , events of Ω (with P(F ∩B) > 0), show that Q(E|F ) = P(E|F ∩B). 30. Another Proof Let A,B,C Ω with P(B) > 0, P(B ∩ C) > 0, and P(B ∩ Cc) > 0. Show that P(A|B) = P(A|B ∩ C)P(C|B) + P(A|B ∩ Cc)P(Cc|B). 31. Married on Campus Suppose that 40% of the students on a campus, who are married to students on the same campus, are female. Moreover, suppose that 30% of those who are married, but not to students at this campus, are also female. If one third of the married students on this campus are married to other students on this campus, what is the probability that a randomly selected married student from this campus is a woman 32. Batteries Suppose that 10 good and three dead batteries are mixed up. Jack tests them one by one, at random and without replacement. But before testing the fifth battery he realizes that he does not remember whether the first one tested is good or is dead. All he remembers is that the last three that were tested were all good. What is the probability that the first one is also good 33. Multiple Choice On a multiple choice exam with four choices for each question, a student either knows the answer to a question or marks it at random. If the probability that he or she knows the answers is 2/3, what is the probability that an answer that was marked correctly was not marked randomly 5 34. Cards A stack of cards consists of six red and five blue cards. A second stack of cards consists of nine red cards. A stack is selected at random and three of its cards are drawn. If all of them are red, what is the probability that the first stack was selected 35. More Cards There are three identical cards that differ only in color. Both sides of one are black, both sides of the second one are red, and one side of the third card is black and its other side is red. These cards are mixed up and one of them is selected at random. If the upper side of this card is red, what is the probability that its other side is black 36. Horses There are two stables on a farm, one that houses 20 horses and 13 mules, the other with 25 horses and eight mules. Without any pattern, animals occasionally leave their stables and then return to their stables. Suppose that during a period when all the animals are in their stables, a horse comes out of a stable and then returns. What is the probability that the next animal coming out of the same stable will also be a horse 37. Chips An urn contains five red and three blue chips. Suppose that four of these chips are selected at random and transferred to a second urn, which was originally empty. If a random chip from this second urn is blue, what is the probability that two red and two blue chips were transferred from the first urn to the second urn 6 MAST20004 Probability Semester 1, 2021 Problem Sheet 3 Random Variables and Distribution Functions 38. More Chips From an urn that contains five red, five white, and five blue chips, we draw two chips at random. For each blue chip we win $1, for each white chip we win $2, but for each red chip we lose $3. If X represents the amount that we either win or we lose, what are the possible values of X and probabilities associated with them 39. Plastic Die The side measurement of a plastic die, manufactured by factory A, is a random number between 1 and 1.25 centimetres. What is the probability that the volume of a randomly selected die manufactured by this company is greater than 1.424 Assume that the die will always be a cube. 40. Families From families with three children a family is chosen at random. Let X be the number of girls in the family. Calculate and sketch the distribution function of X. Assume that in a three-child family all gender distributions are equally probable. 41. Distribution Functions Determine if the following are distribution functions. (a) FX(t) = 0, t < 0 t 1 + t , t ≥ 0. (b) FX(t) = et 2 , t < 0 1 3e t 4 , t ≥ 0. 7 42. Random Points Let X be a randomly selected point from the interval (0, 3). What is the probability that X2 5X + 6 > 0 43. New Car Let the time until a new car breaks down be denoted by X, and let Y = X, if X ≤ 55, if X > 5. Then Y is the life of the car, if it lasts less than 5 years, and is 5 if it lasts longer than 5 years. Calculate the distribution function of Y , FY , in terms of FX , the distribution function of X. 44. Minimum Face Value In the experiment of rolling a fair die twice, let X be the minimum of the two numbers obtained. Determine the probability mass function and the distribution function of X, and sketch their graphs. 45. Probability Mass Function The distribution function of a random variable X is given by FX(x) = 0, x < 2 1/2, 2 ≤ x < 2 3/5, 2 ≤ x < 4 8/9, 4 ≤ x < 6 1, x ≥ 6. Determine the probability mass function of X and sketch its graph. 46. Jury From 18 potential women jurors and 28 potential men jurors, a jury of 12 is chosen at random. Let X be the number of women selected. Find the probability mass function of X. 47. First Six In successive rolls of a fair die, let X be the number of rolls until the first 6 appears. Determine the probability mass function and the distribution function of X. 8 48. More Probability Mass Functions For each of the following, determine the value(s) of k for which p is a probability mass function. Note that in Parts (d) and (e), n is a positive integer. (a) p(x) = kx, x = 1, 2, 3, 4, 5 ; (b) p(x) = k(1 + x)2, x = 2, 0, 1, 2 ; (c) p(x) = k(1/9)x, x = 1, 2, 3, . . . ; (d) p(x) = kx, x = 1, 2, 3, . . . , n ; (e) p(x) = kx2, x = 1, 2, 3, . . . , n. Hint : Recall that n∑ i=1 i = n(n+ 1)2 , n∑ i=1 i2 = n(n+ 1)(2n+ 1)6 . 49. Soap Opera The distribution function for the duration of a certain soap opera (in tens of hours) is F (x) = 1 16 x2 , x ≥ 4 0, x < 4. (a) Calculate f , the probability density function of the soap opera. (b) Sketch the graphs of F and f . (c) What is the probability that the soap opera takes at most 50 hours At least 60 hours Between 50 and 70 hours Between 10 and 35 hours 50. Tyres The lifetime of a tyre selected randomly from a used tyre shop is 10,000X kilometres, where X is a random variable with the density function f(x) = 2 x2 , 1 < x < 2 0, elsewhere. (a) What percentage of the tyres of this shop last fewer than 15,000 kilometres (b) What percentage of those having lifetimes fewer than 15,000 kilometres last between 1
0,000 and 12,500 kilometres 51. Another Density Function Let X be a continuous random variable with density and distribution functions f and F , respectively. Assuming that α ∈ R is a point at which P(X ≤ α) < 1, prove that h(x) = f(x) 1 F (α) , x ≥ α 0, x < α is also a probability density function. 9 52. Investment Suppose that the loss in a certain investment, in thousands of dollars, is a continuous random variable X that has a density function of the form f(x) = k(2x 3x 2), 1 < x < 0 0, elsewhere. (a) Calculate the value of k. (b) Find the probability that the loss is at most $500. 53. Convex Combination Prove that if f and g are two probability density functions, then for α ≥ 0, β ≥ 0, and α+β = 1, αf + βg is also a probability density function. 54. Fuses A box contains 20 fuses, of which five are defective. What is the expected number of defective items among three fuses selected randomly 55. Nonexistant Expected Value It is well known that ∑∞x=1 1/x2 = pi2/6. (a) Show that p(x) = 6/(pix)2, x = 1, 2, 3, . . . is the probability mass function of a random variable X. (b) Prove that E(X) does not exist. 10 56. Distribution Function The distribution function of a random variable X is given by FX(x) = 0, x < 3 3/8, 3 ≤ x < 0 1/2, 0 ≤ x < 3 3/4, 3 ≤ x < 4 1, x ≥ 4. Calculate E(X), E(X2 2|X|), and E(X|X|). 57. Children A newly married couple decides to continue having children until they have one of each sex. If the events of having a boy and a girl are independent and equiprobable, how many children should this couple expect Hint : Note that ∑∞i=1 iri = r/(1 r)2, |r| < 1. 58. Probability Mass Function (a) Show that p(n) = 1 n(n+ 1) , n ≥ 1, is a probability mass function. (b) Let X be a random variable with probability mass function p given in Part (a). Find E(X). 59. Variance Find the variance and the standard deviation of a random variable X with distribution function F (x) = 0, x < 3 3/8, 3 ≤ x < 0 3/4, 0 ≤ x < 6 1, x ≥ 6. 60. Random Integer Let X be a random integer from the set {1, 2, . . . , N}. Find E(X), V(X), and σX . 61. Another Variance Suppose that X is a discrete random variable with E(X) = 1 and E(X(X 2)) = 3. Find V( 3X + 5). 11 62. Soap Opera The distribution function for the duration of a certain soap opera (in tens of hours) is F (x) = 1 16 x2 , x ≥ 4 0, x < 4. (a) Calculate E(X). (b) Show that V(X) does not exist. 63. Aptitude Test The time it takes for a student to finish an aptitude test (in hours) has the density function f(x) = 6(x 1)(2 x), 1 < x < 20, otherwise. Determine the mean and standard deviation of the time it takes for a randomly selected student to finish the aptitude test. 64. Expected Value A random variable X has the density function f(x) = 3e 3x, x ≥ 0 0, x < 0. Calculate E(eX). 65. Computer Network Let Y be a continuous random variable with probability distribution function F (y) = e k(α y)/A, y ≤ α 1, y > α, where A, k, and α are positive constants. (Such distribution functions arise in the study of local computer network performance.) Find E(Y ). 66. Logarithm of a Random Variable Let X be a continuous random variable with probability density function f(x) = 2/x 2, 1 < x < 2 0, elsewhere. Find E(logX). 12 67. Double Exponential Distribution Let X be a continuous random variable with probability density function f(x) = e |x| 2 , ∞ < x <∞. Calculate V(X). 68. Absolute Moment Let X be a discrete random variable. The absolute moment of X of order t is E(|X|t). If 0 < s < r, show that if the absolute moment of order r of X exists, then the absolute moment of order s also exists. 69. Recursive Moments Let X be a continuous random variable with the probability density function f(x) = x sin x pi , 0 < x < pi 0, otherwise. Prove that E(Xn+1) + (n+ 1)(n+ 2)E(Xn 1) = pin+1. 70. Computing Moments using Tail Probabilities If P(X ≤ 0) = 1 and E[|X|n] <∞, show that for n > 0, E[Xn] = n ∫ 0 ∞ xn 1FX(x)dx. 71. Post Office Suppose that X, the interarrival time between two customers entering a certain post office, satisfies P(X > t) = αe λt + βe μt, t ≥ 0, where α + β = 1, α, β ≥ 0, λ, μ > 0. Calculate (a) E[X] ; (b) V(X). 72. Bounded Random Variable Let X be a continuous random variable with set of possible values {x : 0 < x < α} (where α > 0) and distribution function FX . Using integration by parts, prove that E[X] = ∫ α 0 [1 FX(x)] dx. 13 MAST20004 Probability Semester 1, 2021 Problem Sheet 4 Special Probability Distributions 73. Number Plates In a state where license plates contain six digits, what is the probability that the license number of a randomly selected car has two 9s Assume that each digit of the license number is randomly selected from {0, 1, . . . , 9}. 74. Nails A manufacturer of nails claims that only 3% of its nails are defective. A random sample of 24 nails is selected, and it is found that two of them are defective. Is it fair to reject the manufacturer’s claim based on this observation (Hint : assuming 3% of its nails are defective, is the probability of getting a sample of at least 2 defectives out of 24 small ) 75. Five Points From the interval (0, 1), five points are selected at random and independently. What is the probability that (a) at least two of them are less than 1/3 ; (b) the first decimal point of exactly two of them is 3 76. Rare Blood Type A certain rare blood type can be found in only 0.05% of people. If the population of a randomly selected group is 3000, what is the probability that at least two persons in the group have this rare blood type 77. Children A woman and her husband want to have at least a 95% chance for at least one boy and at least one girl. What is the minimum number of children that they should plan to have Assume that the events that a child is a girl and a boy are equiprobable and independent of the gender of other children born in the family. 14 78. Chuck-a-Luck A game often played in carnivals and gambling houses is called chuck-a-luck, where a player bets on any number 1 through 6. Then three fair dice are tossed. If one, two, or all three land the same number as the player’s, then he or she receives one, two, or three times the original stake plus his or her original bet, respectively. Otherwise, the player loses his or her stake. Let X be the net gain of the player per unit of stake. First find the probability mass function of X ; then determine the expected amount that the player will lose per unit of stake. 79. Aircraft Engines Suppose that an aircraft engine will fail in flight with probability 1 p independently of the plane’s other engines. Also suppose that a plane can complete the journey successfully if at least half of its engines do not fail. Is it true that a four-engine plane is always preferable to a two-engine plane Explain. 80. Even Number of Successes What is the probability of an even number of successes in n independent Bernoulli trials Hint : Let rn be the probability of an even number of successes in n Bernoulli trials. By conditioning on the first trial and using the law of total probability, show that for n ≥ 1, rn = p(1 rn 1) + (1 p)rn 1. Then prove that rn = 12 (1 + (1 2p)n). 81. Targets The probability is p that Marty hits target M when he fires at it. The probability is q that Alvie hits target A when he fires at it. Marty and Alvie fire one shot each at their targets. If both of them hit their targets, they stop ; otherwise, they will continue. (a) What is the probability that they stop after each has fired r times (b) What is the expected value of the number of turns taken before the turn where they both hit the target 82. Basketball A certain basketball player makes a foul shot with probability 0.45. What is the probability that (a) his first basket occurs on the sixth shot (b) his first and second baskets occur on his fourth and eighth shots, respectively 83. Light Bulbs The probability is p that a randomly chosen light bulb is defective. We screw a bulb into a lamp and switch on the current. If the bulb works, we stop ; otherwise, we try another and continue until a good bulb is found. What is the probability that at least n bulbs were defective 15 84. Bridge On average, how many games of bridge are necessary until a player is dealt three aces A bridge hand is 13 randomly selected cards from an ordinary deck of 52 cards. 85. Senior Citizens Suppose that 15% of the population of a town are senior citizens. Let X be the number of nonsenior citizens who enter a mall before the tenth senior citizen arrives. Find the probability mass function of X. Assume that each customer who enters the mall is a random person from the entire population. 86. Geometric Distribution Suppose X d= G(p). Calculate the probability that X is even. 87. Negative Binomial For r = 1, 2, . . ., show that, ( r z ) = ( 1)z ( z + r 1 r 1 ) , using the extended binomial coefficient definition on Slide 166. 88. Negative Binomial Distribution For r ∈ (0,∞) and 0 < p ≤ 1, let Z d= Nb(r, p). For z = 0, 1, 2 . . ., show that pZ(z) = ( r z ) pr(p 1)z ≥ 0. 89. Professors’ Cars Of the 28 professors in a certain department, 18 drive foreign and 10 drive domestic cars. If five of these professors are selected at random, what is the probability that at least three of them drive foreign cars 90. Bernoulli Trials Suppose that independent Bernoulli trials with parameter p are performed successively. Let N be the number of trials until x successes, and X be the number of successes in the first n trials. Show that P(N = n) = x n P(X = x). Remark : By this relation, in coin tossing, for example, we can state that the probability of getting a fifth head on the seventh toss is 5/7 of the probability of five heads in seven tosses. 91. Charity In an annual charity drive, 35% of a population of 560 make contributions. If, in a statistical survey, 15 people are selected at random and without replacement, what is the probability that at least two persons have contributed 16 92. Defective Items The policy of the quality control division of a certain corporation is to reject a shipment if more than 5% of its items are defective. A shipment of 500 items is received, 30 of them are randomly tested, and two have been found defective. Should that shipment be rejected That is, assuming 5% of items are defective, is the probability of getting a sample of at least two defectives small 93. Trout To estimate the number of trout in a lake, we caught 50 trout, tagged and returned them. Later we caught 50 trout and found that four of them were tagged. From this experiment estimate n, the total number of trout in the lake. Hint : Let pn be the probability of four tagged trout among the 50 trout caught. Find the value of n that maximizes pn. 94. Annual Income Suppose that 3% of the families in a large city have an annual income of over $60,000. What is the probability that, of 60 random families, at most three have an annual income of over $60,000 95. Misprints Misprints in a particular book occur independently, and on average, there are three misprints in every 10 pages. If every chapter of the book contains 35 pages, what is the probability that Chapters 1 and 5 have 10 misprints each 96. No Crimes In a certain town, crimes occur at a Poisson rate of five per month. What is the probability of having exactly two months (not necessarily consecutive) with no crimes during the next year 97. Bookstore Customers arrive at a bookstore at a Poisson rate of six per hour. Given that the store opens at 9 :30am, what is the probability that exactly one customer arrives by 10 :00am and 10 customers by noon 98. Lottery Tickets Suppose that in Melbourne, on a certain day, N lottery tickets are sold and M win, with M/N very small. To have a probability of at least α of at least one win on that day, approximately how many tickets should be purchased 17 99. Poisson Distribution Let X be a Poisson random variable with parameter λ. Show that the maximum of P(X = i) occurs at λ , where λ is the greatest integer less than or equal to λ. Hint : Let pX be the probability mass function of X. Prove that pX(i) = λ i pX(i 1). Use this to find the values of i at which p is increasing and the values of i at which it is decreasing. 100. Line Segment A point is selected at random on a line segment of length l. What is the probability that (a) the longer segment is at least twice as long as the shorter segment (b) What is the probability that none of the two segments is smaller than l/3 101. Random Angle Let Θ be a random number between pi/2 and pi/2. Find the probability density function of X = tan Θ 102. Expectation of a Logarithm Let X be a uniform random variable over the interval (0, 1). Calculate E( logX). 103. Chicken Pen A farmer who has two pieces of lumber of lengths a and b (a < b) decides to build a pen in the shape of a triangle for his chickens. He sends his foolish son out to cut the longer piece and the boy, without taking any thought as to the ultimate purpose, makes a cut on the lumber of length b, at a point selected randomly. What are the chances that the two resulting pieces and the piece of length a can be used to form a triangular pen Hint : Three segments form a triangle if and only if the length of any one of them is less than the sum of the lengths of the remaining two. 104. Heart Attack The time between the first and second heart attacks for a certain group of people is an expo- nential random variable. If 50% of those who have had a heart attack will have another one within the next five years, what is the probability that a person who had one heart attack five years ago will not have another one in the next five years 105. Exponential Random Variable Let X be an exponential random variable with parameter λ. Find P (|X E(X)| ≥ 2σX) . 18 106. Telephone Call Mr. Jones is waiting to make a phone call at a train station. There are two public telephone booths next to each other, occupied by two persons, say A and B. If the duration of each telephone call is an exponential random variable with λ = 1/8, what is the probability that among Mr. Jones, A, and B, Mr. Jones will not be the last to finish his call 107. Double Exponential Random Variable The random variable X is called double exponentially distributed if its density function is given by f(x) = ce |x|, ∞ < x <∞. (a) Find the value of c ; (b) Prove that E(X2n) = (2n)! and E(X2n+1) = 0. 108. Radio Tubes Let X, the lifetime (in years) of a radio tube, be exponentially distributed with mean 1/λ. Prove that X , the integer part of X, which is the complete number of years that the tube works, is a geometric random variable. 109. Gamma Distribution Let X d= γ(r, α). If c > 0, show that cX d= γ(r, α/c). 110. Defective Light Bulbs A manufacturer produces light bulbs at a Poisson rate of 200 per hour. The probability that a light bulb is defective is 0.015. During production, the light bulbs are tested one by one, and the defective ones are put in a special can that holds up to a maximum of 25 light bulbs. On average, how long does it take until the can is filled 111. Born on Christmas Day A small college has 1095 students. What is the approximate probability that more than five students were born on Christmas day Assume that the birthrates are constant throughout the year and that each year has 365 days. 112. Beta Mean and Variance Let X d= Beta(α, β). Calculate E[X] and V(X). 19 113. Beta Random Variable For which value of c is the following a probability density function of some random variable X Find E[X] and V(X). f(x) = cx 4(1 x)5, 0 < x < 1 0, otherwise. 114. Blood Pressure Medicines Suppose that new blood pressure medicines introduced are effective on 100P% of the patients, where P is beta random variable with parameters α = 20 and β = 13. What is the probability that a new blood pressure medicine is effective on at least 60% of the hypertensive population 115. Resistors The proportion of resistors a procurement office of an engineering firm orders every month, from a specific vendor, is a beta random variable with mean 1/3 and variance 1/18. What is the probability that next month, the procurement office orders at least 7/12 of its purchase from the vendor 116. Pareto Random Va