Probability and Statistics - Problem Set c Keith M. Chugg October 2, 2015 1 Preliminaries, Combinatorics, Set Probability 1.1. A number of bats are in a cave. 2 bats can see out of their left eye. 3 bats can see out of the right eye. At least 5 bats cannot see out of their left eye. At least 4 bats cannot see out of their right eye. What is the smallest number of bats that can be in the cave? 1.2. Let the sample space associated with a certain random experiment be U = {a, b, c, d, e}. You are interested in a probability model which will allow you to dene the probability of the sets {a} and {b, c, d}. Dene the smallest -algebra of events which allows the probability of these sets to be measured. 1.3. How many events are in the smallest sigma algebra that contains the two events A1 and A2 which form a partition of the sample space? 1.4. The Union Bound: i = 1, 2 . . . n. (a) P (A (b) P (A (c) P ( Prove the following results for arbitrary events A, B, C and Ai for B) P (A) + P (B). When does equality hold? B C) P (A) + P (B) + P (C). n i=1 Ai ) n i=1 P (Ai ). 1.5. Show that if A B then P (A) P (B). 1.6. Two fair dice are rolled and the sum of the outcome is noted. A student declares that the probability that a 7 is rolled is 1/11. The reasoning is that there are 11 possible outcomes and only one results in a 7. Is this correct? Explain. If this is incorrect, provide the correct answer. 1.7. How many distinct permutations of 4 red balls, 2 white balls and 3 blue balls are possible? 1 c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 2 1.8. A probability model for the USC football team's season is that each game is won or lost independently of the other games. The probability that a given game is won is 0.9. Given that USC has won their rst 3 games, what is the probability that they will win all of their remaining 10 games? 1.9. A binary word is made up of 8 bits, each taking on the values 0 or 1. If a binary word is selected at random, what is the probability that it will have exactly three 1's? What is the probability that it will have fewer than four 1's? 1.10. A state lottery is played by picking six numbers in the range {1, 2 . . . 49} - each of the numbers should be dierent. The state then draws 6 balls at random from an urn containing 49 balls labeled from 1 to 49 without replacement. What is the probability of winning (getting all six numbers) with one ticket? 1.11. A hot dog vendor provides onions, relish, mustard, catsup, and hot peppers for your hot dog. Determine how many combinations of toppings (no \"double\" toppings) are possible assuming: (a) You use exactly one topping. (b) You use exactly two toppings. (c) There are no restrictions on how many toppings you can use. 1.12. There are m teams in league A and n teams in league B. On game day there are k games played (k n and k m); each game pits a team from league A against a team from league B. A given team can play at most once on game day. How many dierent sets of match-ups (\"schedules\") can be made for game day? (a) If a '\"schedule\" includes the teams playing and a distinct time for each of the k games? (b) If a \"schedule\" includes only the team match-ups and no information on time? 1.13. Consider the set of 4-tuples (w, x, y, z) where w, x, y, and z are non-negative integers and w + x + y + z = 32 If a member of this set is drawn at random, what is the probability that the value of w in the drawn 4-tuple is 4? 1.14. There are 10 problems on a TRUE/FALSE exam. You ll out the exam randomly. (a) What is the probability that you answer all 10 questions correctly? (b) What is the probability that you answer all 10 questions incorrectly? (c) A passing grade is 7 or more correct. What is the probability that you pass? Hint: The events of 7, 8, 9 and 10 correct are mutually exclusive. 1.15. There are 100 computers in a given production run. The QA engineer suggests that, in practice, not all need to be tested. She argues that with high probability a bad lot can be detected by testing only M < 100 of the computers. To check this claim, consider the following: c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 3 (a) There are D defective computers among the 100; T of the 100 computers are selected at random and tested. What is the probability that F of the T tested fail? (b) An entire lot should be rejected when 20 or more computers are defective. The following test is suggested: Randomly select T computers and test each; if more than B fail, the lot is rejected. What is the minimum value of T and a corresponding threshold B which ensures that bad lots will be rejected with probability greater than 0.9 and that lots containing 0 or 1 defects will be rejected with probability no more than 0.10? (You may need a computer to determine this answer). 1.16. Population Estimation: It is desired to estimate the number of foxes in a forest without catching all of them. Previously, 10 foxes have been captured, tagged and released. A month later 20 foxes are captured and 5 of these have tags. (a) If the actual number of foxes in the forest is N (unknown to us), what is the probability of this event as a function of N ? Denote this probability by p(N ). (b) The estimate of the number of foxes is taken to be the value of N which maximizes p(N ). What is this estimate? Hint: Compare p(N )/p(N 1) against 1 to perform the maximization. 1.17. The state lottery was recently changed to decrease the probability of winning the grand prize. The new game is played by selecting a group of 6 numbers from {1, 2, 3 . . . 51} (two more than previously). The state selects a group of 6 numbers from {1, 2, 3 . . . 51} and you win the grand prize if all 6 of your numbers match the state's. The probability of winning became so low for the new game that people began to try dierent ways of \"increasing their odds.\" (a) Let p denote the probability that a given ticket wins. Determine p for the new game. (b) (The Determined Individual) Joan decides that no matter what the probability of winning is, she will eventually win if she plays enough times. She buys 1 ticket every 24 hours of every 365 days per year for 80 years of her life. Each ticket costs $5 and she selects her numbers independently from one ticket to the next. Determine: i. Pr {Joan wins the Lottery at least one time} ii. Amount of money she spends trying (c) (The Group Eort) Wayne decides that he would be happy splitting the grand prize with his friends if he can increase his chance of winning. Wayne organizes N of his friends and each (independently) buys exactly one ticket for this week's drawing. The minimum value of N (Nmin ( )) to insure that the probability that there is at least one winner among the group is : Nmin ( ). Calculate the value of Nmin ( ) for = 105 , = 103 , = 0.1, = 0.5. (d) If Wayne has gathered enough friends to win with probability 0.5 using the above strategy, then can you gure out a way that they can increase their win probability above 0.5 without adding more people to the group? (e) (The Compromising Bureaucrat) The state lottery commissioner realizes that, because of the tougher odds, sales are slumping since the the game was changed. Her remedy is to introduce a consolation prize: if exactly 3 of your 6 numbers match any 3 of the c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 4 state's 6, then you win the consolation prize. Determine the probability that a single ticket wins the consolation prize. 1.18. Every person has been assigned a unique (i.e no two people have the same) ID number of the form x9 x8 x7 x6 x5 x4 x3 x2 x1 , where xi {0, 1, 2 . . . 9} for i = 1, 2 . . . 9. A class consists of M students selected at random from the general population. For compactness, the instructor keeps his records based only on the shortened student ID of x4 x3 x2 x1 . Let the size of the general population be n, and assume that n is also equal to the number of possible (long) ID's. Determine n and the probability that the instructor will have a conict in his record (i.e. a conict will occur if there are at least two students in the class with the same short student ID). 1.19. There are 15 pizza toppings available. You can order any of these toppings either as standard (i.e., a single amount) or as extra (i.e., a double amount). In other words, there are single and double toppings allowed, but no triple, quadruple, etc. How many dierent 4-topping pizzas are possible? If one 4 topping pizza of this type is selected at random, what is the probability that it has a double topping of at least one item? 1.20. Amy sees 3 quarters, 2 dimes, 1 nickel, and 3 pennies on a table. She will take some of this change and place it in her pocket. Coins of a given amount are not distinguishable. (a) How many dierent (non-empty) combinations of coins can Amy take? (b) How many dierent (non-zero) amounts of money can Amy take? 1.21. Jim has 20 identical shirts and 4 drawers in which to place them. How many ways can he do this? 1.22. Let A and B be given events with, P (A) > 0, P (B) (0, 1) and P (A|B) all known. Determine P (A|B c ) in terms of these known quantities. If, in addition, A and B are mutually exclusive, what is the relationship between P (A) and P (A|B c )? 1.23. Suppose that A B, P (A) = 1/4 and P (B) = 1/3. Determine P (A|B) and P (B|A). 1.24. Consider three events A, C, and D with P (A|C) = 1/2, P (A|D) = 1/5, P (C) = 1/5, and P (D) = 2/5. Also, events C and D are mutually exclusive. Provide an expression for P (A|C D) in terms of the quantities provided. Then, provide the numerical value for this conditional probability. 1.25. Consider a random experiment in which a student is selected at random from the USC student body. Let A be the event that a student is older than 20. Let B be the event that a student is male. Let C be the event that a student is an engineering major. Determine the set relations for the following events: (a) A student is male or older than 20. (b) A student is female and 20 or younger. (c) A student is female, an engineer, and 20 or younger. (d) A student is not male and not older than 20, but is an engineer. c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 5 (e) A student is an engineer or a female 20 or younger. 1.26. For the following events A and B, state whether they are independent, mutually exclusive or neither. (a) A = ight 1712 departs LAX on time on 9/6/05, B = ight 1712 arrives in Denver on time on 9/6/05. (b) A = a given person is a Democrat, B = a given person is a Republican. (c) A = a given adult is over 6 feet tall, B = a given adult has IQ greater than 120. (d) A = OPEC embargo, B = reduction is U.S. gasoline prices by more than 50% (e) A = New England Patriots win Pro Football championship, B = USC Trojans win College Football Championship. 1.27. Two fair dice are rolled; conditioned on the event that the dice land on dierent numbers, what is the probability that at least one die lands on 6? 1.28. There are 10 problems on a TRUE/FALSE exam. 20% of the students are completely unprepared for the exam and answer randomly. The remaining students are prepared. Prepared students pass the exam (7 or more correct) with probability 0.95. (a) What is the probability that a randomly selected student passes? (b) Given that a student does not pass, what is the probability that he was prepared? 1.29. The Rose Bowl is played between the champions of the PAC-10 and BIG-10 conferences. USC is the PAC-10 champion with probability 0.6 and they win the rose bowl 90% of the times that they represent the PAC-10. UCLA wins the PAC-10 20% of the time and given that they make it to the Rose Bowl, they win it with probability 0.2. When one of the other PAC-10 teams win the conference championship, they are equally likely to win or lose in the Rose Bowl. There are 10 teams in the Pac-10. (a) What is the probability that USC wins the Rose Bowl? (b) What is the probability that the Big-10 representative wins the Rose Bowl. (assume that Rose Bowl ties occur with zero probability). (c) Given that the BIG-10 wins the Rose Bowl in a certain year, what is the probability that UCLA represented the PAC-10? Under the same condition what is the probability that USC represented the PAC-10? 1.30. A study is being conducted in an attempt to correlate quality of education and income level. Engineers who have been in the workforce for at least ten years are categorized according to their income level being one of HIGH, MEDIUM, or LOW. Their undergraduate colleges are also noted and this population is limited to those who graduated from USC, Stanford, or Harvey Mudd College (HMC). Half of this population graduated from USC, ten percent from HMC and the rest graduated from Stanford. For USC graduates, 60% are HIGH income earners and 30% are MEDIUM income earners. Half of Stanford graduates earn HIGH income and 20% earn LOW income. HMC graduates make the most; 90% of HMC graduates earn high income and the rest are equally likely to be LOW or MEDIUM wage earners. c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 6 (a) What is the probability that an engineer is an HMC graduate and makes a HIGH salary? (b) What is the probability that an engineer is an HMC graduate and does not make a LOW salary? (c) Determine the probability that an engineer has a LOW, MEDIUM, or HIGH income: i. P (LOW income) ii. P (MEDIUM income) iii. P (HIGH income) (d) If an engineer has a LOW income, what is the probability that she graduated from Stanford? (e) Suppose that you're given that a particular engineer has a HIGH income and asked to make your best decision as to which school she attended. Describe a strategy for making this decision and then use your strategy to make your best decision. 1.31. A computer memory chip fails between times t1 and t2 with probability t2 P (Fails between t1 and t2 ) = ez dz, (1) t1 where t1 and t2 are measured in units of hours after start-up and is a constant with units of (hours)1 . (a) What is the probability that the chip does not fail in the rst T hours? (b) What is the probability that the chip does fail in the rst T hours? (c) Given that the chip has not failed in the rst S hours, what is the probability that it will fail between time S and S + T ? 1.32. A fair coin is ipped 106 times, determine: Pr 499, 000 # of \"heads\" in 106 ips 501, 000 1.33. A simple method for detecting errors in a binary digital communications system is to use a parity check bit. A packet consists of (n 1) data bits and 1 parity bit. The parity bit is selected so that an even number of \"1's\" contained in the transmitted packet of length n. The signal is then distorted and the receiver makes errors independently at each bit location with probability p. The number of 1's in the detected signal is then counted; if this number is even the packet is labeled good, otherwise it is labeled bad and the data is ignored. (a) What is the probability that a packet is declared bad? (b) Derive upper and lower bounds for the probability in (a) which can be made arbitrarily tight by including more terms. (c) Use the family of bounds found in (b) to obtain a numerical answer for the probability of declaring the packet bad when n = 1000 and p = 5 103 . Repeat for n = 1000 and p = 1 104 . c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 7 1.34. The hypergeometric probability law governs the selection of k items from a set of n total where m of the n are 'marked'. For a specic example, suppose there are n people in a room, m of whom are left-handed (the rest are right-handed). If k people are drawn without order and without replacement, then the probability of having l left-handed people is Pr {l left-handers in k draws} = m l n k nm kl , lk (a) Show that if the ratio m/n is xed that as n becomes large, this tends toward the binomial distribution with p = m/n. (b) Consider a community of n people where 20% of the people are left-handed. Suppose a panel of 3 people is formed from this community by selecting at random. What is the probability that there will be no left-handed people on the panel? Do this for n = 10 and repeat for n = 100. (c) Compare the large n approximation from the rst part of this problem to the exact answer obtained in the second part. What is the intuitive reasoning why the binomial distribution approximates the hypergeometric distribution for large n? 1.35. Ms. Nev R. Passem is a new QA engineer for a large computer manufacturer. Her rst task on the job is to review the company's \"burn-in\" testing procedures. She is told that there are two types of computers: good and bad. The company wants to sell the good computers and use bad computers for salvage. The computers fail between times t1 and t2 (both 0) with probability t2 Good Computers: Bad Computers: Pr {fail in (t1 , t2 ]} = Pr {fail in (t1 , t2 ]} = t1 t2 g eg x dx b eb x dx. t1 The company's current test is to run the computers from time 0 to T , and if a given computer does not fail during this burn-in period it is accepted and shipped; if it fails it is rejected and used for salvage. She is also told that the company is concerned with two measures of quality Detection Probability: False Alarm Probability: PD = Pr {rejecting a computer given that it is bad} PF A = Pr {rejecting a computer given that it is good}. The company's quality standards are PF A 0.01 and PD 0.99. (a) Determine the False Alarm and Detection probabilities as a function of T ,b ,g . (b) Nev is not sure that PD and PF A are the best standards to use; instead she suggests determining the probability that a rejected computer is good or bad. She gures that the probability that a given computer is bad is 0.01. The company tells her that their current test meets the standards with PF A = 0.008 and PD = 0.993. Using the results of (a), determine Nev's new standards: c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 8 i. Pr {Computer is bad|rejected} ii. Pr {Computer is good|rejected} (c) The company is interested in adjusting the test duration and they have asked Nev to determine the eects of varying T . Ms. Passem is told that g = 1/600 and b = 1 in units of (hours)1 . Nev realizes that, in order to achieve the company's standards (PF A 0.01 and PD 0.99) with the given parameters, the test duration must be in a specic interval. Determine the minimum and maximum values of T (Tmin T Tmax ) so that the company's standards are met. (d) After performing the above calculations, Nev is concerned that if the company's values for g and b are inaccurate, then designing a test to meet the standards (PF A 0.01 and PD 0.99) will be impossible. She determines that the ability of a test of this type (with any choice of T ) to distinguish between good and bad computers is a function of = b . g Determine the minimum value of (min ) which allows a test of this type to meet the false alarm and detection probability requirements. 1.36. Bill is being interviewed and is given one question that will determine whether he is hired. The interviewer brings two buckets into the room, bucket 1 and bucket 2, along with 50 green balls and 50 red balls. The interviewer tells Bill that she will leave the room and he is to put all 100 balls into the buckets such that neither bucket is empty. She will return to the room, randomly select a bucket and then draw a ball from that bucket. If the drawn ball is green, Bill is hired, if it is red, he is not hired. (a) If Bill places r red balls and g green balls into bucket 1, what is the probability that he will be hired? (b) What strategy should Bill take to maximize his chance of being hired - i.e., what are the best choices for r and g? 1.37. Use a computer (e.g., the program binomial.c, the spreadsheet, Matlab, etc.) to nd example n, p and k where the following conditions hold: (a) Both the Poisson and Gaussian approximations are valid. (b) The Poisson approximation is valid, but the Gaussian approximation is invalid. (c) The Gaussian approximation is valid, but the Poisson approximation is invalid. (d) Both approximations are invalid. (e) For at least one case where the Gaussian approximation is valid, demonstrate that it may not hold far from kmax . 1.38. Suppose that each child born to a couple is equally probable to be a girl or a boy. Also assume births are independent of the sex distribution of the previous children. For a couple having 6 children, determine the probability of the following events: (a) All children are the same sex. c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 9 (b) The 3 eldest are boys and the 3 youngest are girls. (c) There is at least one girl. (d) The sixth child is a girl given that all 5 of their children are boys. 1.39. A certain loaded die produces a \"1\" with probability 0.5, and all other outcomes with equal probability. This loaded die is placed in a box with seven fair dice. A die is selected at random from the box and rolled 10 times. Determine the probability that the die selected was the loaded die given the follow events: (a) Exactly 5 of the 10 rolls result in 1's. (b) There are three 1's, zero 2's, one 3, two 4's, three 5's and one 6. (c) No rolls come up 2. 1.40. This problem deals with the lottery game dened in problem 1.10 (i.e., 49 pick 6 lottery). Assume that each ticket purchased wins independently of the all other tickets. The lottery is played once per week and 20 million tickets are sold each week. Determine the following: (a) The probability that in a given week there are exactly k winners. Give numerical answers for k = 0, k = 1, and k = 3. (b) The probability that nobody wins for the rst three weeks of March in a specic year. (c) The probability of at least 50 winners in 50 weeks. 1.41. Messages are sent through a satellite communication system. A message is received in error with probability 105 . Errors occur independently for dierent messages. Suppose 100,000 messages are sent across this this channel, nd a good numerical answer for the probability that 4 or fewer errors occur. 1.42. Consider three events A, B, and C where A and C are mutually exclusive. Give the simplest expression for the following probabilities: (a) P (A B C) (b) P (A C|B) (c) P (A|B C) 1.43. Consider the probability space associated with rolling a fair die with sample space U = {1, 2, 3, 4, 5, 6}. Provide an example of two non-empty events associated with this experiment, A and B, that are mutually exclusive. Provide an example of two non-empty events associated with this experiment, C and D, that are statistically independent. 1.44. Consider a class with n students. What is the probability that at least two people in the class share the same birthday? Do not consider the year born or account for leap years. 1.45. A class has 39 students and no large rooms are available for the midterm exam. Instead, the exam will be given in 3 separate rooms. How many ways can the class be split into 3 equal-sized groups for the exam? c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set A B 10 C Figure 1: The relation between events A, B, and C. 1.46. Consider three events A, B, and C with set relations show in Fig. 1. Give the simplest expression for the following probabilities: (a) P (A B C) (b) P (A B C) (c) P (A|B C) (d) P (A|B C) (e) P (B C|A) 1.47. You and two friends drive separately to the movies. The movie parking lot has 20 spaces, all in a row. If 15 spaces are occupied at random, what is the probability that you and your two friends can nd adjacent parking spots? In other words, what is the probability that there are at least 3 adjacent empty spaces? Can you verify your answer via Monte Carlo simulation? Can you generalize this to n spots with m occupied and searching for k in a row? 1.48. Two fair dice are rolled. Let D be the event that the sum of the dots on the two dice is an odd number and let F be the event that at least one die came up 4. (a) Find P (F ), P (D), and P (F |D). Are F and D independent? (b) Suppose you know that a 4 was rolled on at least one die and you are asked to bet whether D or Dc (even sum) occurred. Which would you bet? 1.49. Suppose Los Angeles is made up of 50% USC people and 50% UCLA people. Steve selects 8 people at random from this Los Angeles population to form a focus group. From this group of 8, Sarah selects 4 people at random for another experiment. Let A be the event that of the 4 people selected by Sarah, exactly 2 are USC people. Find the probability of this event. 1.50. An n-bit word is sent across a binary symmetric channel (BSC) with error probability . This means that each bit location in the n-bit word is ipped independently with probability . (a) A word error occurs if any of the n bits is ipped. What is the probability of word error Pw ? c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 11 (b) Reconsider part (a) when an error correcting code is used. Specically, a t-error correcting code is used which will correct any pattern of t or fewer bit ips (i.e., t or fewer channel errors). The code may also correct some error patterns of more than t ips, but not all. Let Pw,decoded denote the probability of word error after decoding the code. Find a good upper bound on this probability. (c) A simple Hamming code has n = 7 and t = 1. Evaluate the uncoded word error probability from (a) and the decoded word error probability from (b) for this code assuming = 0.03. 1.51. You work at a company providing web services and are tasked to write a program to detect a denial of service (DoS) attack. Your company has a test to detect increased trac, so the challenge you are faced with is to quickly distinguish a DoS attack from a burst of heavy user activity (heavy-use). Let Bk be the event that k service requests are received during some prescribed unit of time. Your colleagues have collected data from past DoS attacks and past heavy-use periods so you have a good model for P (Bk |H) and P (Bk |A) where H is the event of valid, heavy user activity and A is the event corresponding to a DoS attack. You also know that valid heavy use periods are 9 times more probable than DoS attacks. (a) Find an expression for the a posteriori probability of a DoS attack and a heavy-use period, given that k service requests were received -i.e., P (A|Bk and P (H|Bk ). (b) Consider a specic model for the number of service requests. Specically, there are n service request opportunities and a request occurs during each of these independently with probability p. When a DoS attack is present, p = pA . During a period of heavy use p = pH . Given that Bk has occurred (k {0, 1, 2 . . . n}) provide a good rule for declaring that a DoS attack has occurred (otherwise a period of heavy-use will be declared). Simplify this rule as much as possible. (c) Consider the rule from part (b) with n = 10, pA = 0.6 and pH = 0.2. Describe your rule from (b) in this case for each value of k from k = 0 to k = 10 - i.e., either \"DoS\" or \"heavy\" is declared for each value of k. 1.52. A team of social scientists has conducted a study of the role that personality plays in teams in the classroom. Specically, they classify each student into one of two personality types: introverts and extroverts. They studied two-student teams drawn at random from a class and drew conclusions about teams comprising 2 introverts, 2 extraverts, or mixed teams. You know that the researchers used just two classes in their study - an engineering class and a business class - and that they selected between these two class randomly. There were 12 students in the business class and 8 of them were extroverts. There were 4 extroverts and 10 introverts in the engineering class. You are curious about the validity of the study, so you decide to put some of your probability knowledge to work. (a) Find the probability of each type of team when the business class is used - i.e., P (2 extraverts|biz), P (2 introverts|biz), and P (1 extravert, 1 introvert|biz). c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set F X(u) 12 (z) 6a 5a 4a 3a 2a a -2 -1 0 1 2 3 z Figure 2: The cdf of X(u). (b) Find the probability of each type of team when the engineering class is used. (c) Given the type of the team, nd the probability it was drawn from the business or engineering class. (d) One of the researchers' main conclusions was that a team of two extroverts will not be eective, while a team of two introverts is highly eective. Do your results above make you question this conclusion? Explain. 2 Random Variables 2.1. Consider the cdf of X(u) shown in Fig. 2. If Pr {X(u) 3} = 1, then determine: (a) a (b) Pr {X(u) 2} (c) Pr {X(u) = 1} (d) Pr {2.5 X(u) < 3} (e) Pr {X(u) = 3} 2.2. A fair coin is tossed 4 times. Determine and sketch the cumulative distribution function for the following random variables: (a) X(u), the number of \"Heads\" observed. (b) Y (u), the number of \"Tails\" observed. (c) D(u) = X(u) Y (u), the dierence between the number of heads and number of tails. c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 13 2.3. The geometric random variable is dened as the number of coin ips required to get the rst \"Head.\" If heads occur with probability p, determine Pr {X(u) = k} and the cdf. 2.4. Show that if X(u) is a Gaussian (also known as \"normal\") random variable with pdf fX(u) (z) = N (z; m, 2 ) = 1 2 2 then Pr {X(u) x} = Q (z m)2 , 2 2 exp xm . (2) (3) 2 2.5. X(u) is a Gaussian random variable with mean mX = 2 and variance X = 25. Determine the probability that X(u) is greater than 10. 2.6. The random variable X(u) has pdf given by fX(u) (z) = cz(1 z) 0 0z1 otherwise (4) (a) Find c. (b) Find Pr {1/2 X(u) 3/4}. (c) Find FX(u) (z). 2.7. The probability mass function (pmf) of X(u) is pX(u) (k) = Pr {X(u) = k} = Ka|k| k = 0, 1, 2 . . . (5) where K > 0 is a constant. Determine the following: (a) Possible range for a (b) K (c) X(u) (z) = E z X(u) (d) The mean of X(u). 2.8. A Rayleigh random variable has cdf FX(u) (z) = 1 ez 0 2 /(2 2 ) z0 z<0 (6) determine the following (a) pdf of x(u). (b) pr { x(u) 2}. (c) {x(u) 3}. 2.9. let be an exponential random variable with parameter . segment real line into 5 equiprobable disjoint intervals. c k.m. chugg - october 2, 2015- probability and statistics problem set 14 a -2 2 x figure 3: considered in 2.15. 2.10. time that jim arrives his oce is modeled as variable, y (u), measured minutes after eight am today. for each cases below fy (u) (z|y> t) where t is a xed number The probability that Jim arrives in the next minute given that he has not arrived at time t: Pr {Y (u) (t, t + 1]|Y (u) > t}. (a) Y (u) is an exponential random variable with parameter in (minutes)1 . (b) Jim arrives sometime between 8 and 9 o'clock, with the probability uniformly distributed in this region. (i.e. Y (u) is uniform on [8, 9)). Let Y (u) model the arrival time in minutes after 8:00. 2.11. Show that the Geometric random variable (see problem 2.3) has the memoryless property: Pr {X(u) m + k|X(u) > m} = Pr {X(u) k}, m, k 0 integers, (7) or FX(u) (m + k|X(u) > m) = FX(u) (k). 2.12. Let X(u) have pdf N (x; 0; 2 ), nd the pdf of X(u) conditioned on the event {X(u) > 0}. 2.13. If X(u) is a Bernoulli random variable, equal to 1 with probability 0.25 and equal to 0 with probability 0.75, determine the mean and variance of X(u) and the probability that X(u) is greater than 0.1. 2.14. The pmf of X(u) is given by 1 pX(u) (k) = Pr {X(u) = k} = 2|k| 3 k = 0, 1, 2 . . . Determine the mean and variance of X(u). 2.15. The random variable X(u) has pdf as shown in Fig. 3, where A is a positive constant. (a) Determine the constant A (b) Find and sketch the cdf of X(u) (c) Determine the probability that X(u) is greater than 1 (d) Determine the mean and variance of X(u) (8) c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 15 (e) Determine E {sin(10X)} 2.16. There are two boxes of light bulbs. A bulb is selected from a box at random and used. The time to failure is an exponential random variable X(u) (conditioned on the failure rate), for light bulbs from either box. The failure rate (in units of 1/hours) is modeled as a random variable R(u). The pdf of R(u) depends on which box the bulb is drawn from Box 1: fR(u) () = Box 2: fR(u) () = 1/M 0 [0, M ) otherwise exp(/500) U(). 500 Determine and sketch the following for the two cases of M = 20 and M = 5: (a) The pdf of the failure rate R(u) (b) The a-posteriori pdf of R(u) given that the selected bulb fails during the rst 10 hours of use. 2.17. Let X(u) be a continuous random variable and Y (u) = [X(u)]k . Find the pdf of Y (u) for k = 1, 2, 3 . . .. 2.18. Let Y (u) = g(X(u)) with 1 1 1 fX(u) (x) = ex U(x) + (x) + (x 2) 2 4 4 2 x < x < 1 g(x) = 1 1x<2 4 x 2 x < . Determine and sketch the pdf of Y (u). 2.19. Consider the random variable X(u) with pdf fX(u) (x) = 1+x 2 0 1 x +1 otherwise If Y (u) = 3X(u) 2, determine and sketch the pdf of Y (u). 2.20. As a newly hired systems engineer, it is your job to analyze the cascaded system shown in Fig. 4. The corresponding denitions are 1 |x| h(y) = ln(y) g(x) = r(z) = 1 z 1/4 0 z < 1/4 Y (u) = g(X(u)) Z(u) = h(Y (u)) W (u) = r(Z(u)). c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set X(u) g(.) Y(u) Z(u) h(.) r(.) 16 W(u) Figure 4: The system considered in problem 2.20. The pdf of the input random variable is fX(u) (x) = 4x3 0 x (0, 1) x (0, 1) (9) Determine: (a) The means of X(u) and Y (u) (b) The mean and pdf of Z(u) (c) The mean and pdf of W (u) 2.21. A \"dead-zone\" nonlinearity is dened as 0 a < x < a g(x) = x a x a x + a x a, where a is a positive constant. Determine and sketch the pdf of Y (u) = g(X(u)) when X(u) 2 is a Gaussian random variable with mean mX and variance X . 2.22. A computer routine called rand(a,b) returns a uniform random number between a and b. If you want to generate a mean zero, unit variance uniform random number, what values of a and b would you use in the function call? 2.23. Let X(u) be an integer valued, non-negative random variable (Pr {X(u) < 0} = 0), with Pr {X(u) = k} = pX(u) (k) for k = 0, 1, 2 . . .. (a) Show that in this case mX = E {X(u)} = k=0 Pr {X(u) > k}. (10) Use this result to nd the mean of the geometric random variable. (b) A similar result holds for the case of a continuous, non-negative random variable: mX = E {X(u)} = 0 (1 FX(u) (x))dx. (11) To see how this can be useful, compute the mean of the exponential r.v. using this formula. c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 17 2.24. Let (u) be a random variable uniformly distributed on [0, 2). Find the mean and variance of the following random variables (a) X(u) = sin (u) (b) Y (u) = cos (u) (c) R(u) = [X(u)]2 + [Y (u)]2 (d) Z(u) = X(u)Y (u). 2.25. USC plays 13 football games this season and wins each, independently, with probability 0.9. What is the expected number of wins for the season? 2 2.26. Let X(u) be a mean zero Gaussian random variable with variance X . Find the expected value of X(u) conditioned on the event {X(u) > 0}. (Hint: see problem 2.12). 2.27. If the random variable (u) is uniformly distributed on the interval from 0 to 2, then X(u) = cos((u)) has arc-cos pdf: fX(u) (x) = 1 2 1x2 0 1 x +1 otherwise Determine the mean and variance of X(u). 2.28. Find the following probabilities: Pr {X(u) > mX } Pr {X(u) mX } 2 for the cases of X(u) Gaussian (variance x ), uniform on [a, b], and exponential with mean 1/. What is a sucient condition for these two probabilities to be equal? 2 2.29. Let X(u) be a Gaussian random variable with mean zero and variance X . Determine: (a) E e3X(u) (b) E e3X(u) (c) E {cosh(3X(u))} 2.30. If the normalized time that a professor requires a Ph.D. student to study before graduating is x, then the number of students studying under the professor may be modeled as a random variable N (u) with Pr {N (u) = k for grad. time x} = xk x e k = 0, 1, 2 . . . k! (12) For a particular professor, Prof. I.M. Tubuzy, the time required for a student to graduate is best modeled as a random variable X(u) with fX(u) (x) = ex U(x), (13) c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 18 where > 0 is a constant. In other words, the probability that Dr. Tubuzy has k students conditioned on his graduating time is Pr {N (u) = k|X(u) = x} = xk x e k = 0, 1, 2 . . . k! (14) (a) Determine the unconditional pmf of N (u). (b) What type of random variable is M (u) = N (u) + 1? Recognizing this, determine the second moment description of N (u). (c) Prof. Tubuzy has n students. Find the pdf and the average value of X(u) conditioned on this information. (d) Determine the condition on n so that E {X(u)|N (u) = n} > E {X(u)} (15) In other words, when does learning that Dr. Tubuzy has n students increase the expected graduating time? 2.31. Find the mean and variance of the geometric random variable dened in problem 2.3. 2.32. Let X(u) be Gaussian with mean m and variance 2 . Find E {|X(u)|}. (Hint: rst nd E {|X(u) m|}. 2.33. The pdf of Y (u) is fY (u) (y) = y exp 2 y 2 2 2 U(y). (16) Determine the mean and third moment of Y (u) 2.34. Using the characteristic function (or moment generating function), nd E [X(u)]4 when 2 X(u) is Gaussian with zero mean and variance X . What is the third moment in this case? 2.35. A fair coin is ipped 100 times. Find a lower bound, based on Chebychev's bound, on the probability of the event that the number of heads observed is in {45, 46, . . . 55}. Compare this to the the exact probability is 0.73. 2.36. Show that, for a > 0, Pr {X(u) > a} Pr {|X(u)| > a}. In the case where the pdf of X(u) is symmetric around zero, show that 1 Pr {X(u) > a} = Pr {|X(u)| > a} 2 2.37. The height of a randomly selected USC student is measured. The average height of a USC student is 65 inches. (a) Determine a good upper bound for Pr {the student is at least 74 inches tall}. c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 19 (b) If, in addition, it is known that the standard deviation of the height is 4 inches, specify a and b so that Pr {student is between a and b inches tall} > 0.9 and give a good lower bound on Pr {the student is between 60 and 72 inches tall}. 2.38. The purpose of this problem is to gain a feel for tail-probability bounds discussed in class. 2 Let the random variable of interest be Gaussian with zero mean and variance x . Develop (if necessary) and investigate the following bounds for Pr {X(u) ax }: Specialized Markov bound: Use the second result from problem 2.36 and the result of problem 2.32. Chebychev bound. Fourth moment bound: Use the results of problem 2.34. Cherno bound. The over bound given in the Q-function handout. The exact value of the probability. Plot these expressions against a. 2.39. Consider a national election between two candidates, one Republican and one Democrat. The voting population is 40% Democrat and 20% Republican with the remaining voters being independent (i.e., neither Republican nor Democrat). Both Republican and Democrat voters vote for their party's candidate with probability 0.9. (a) In this part, consider the probability that an independent voter votes for the Republican candidate to be p. For what range of p will a Republican victory be more probably than a Democrat victory? Assume that p = 0.3. If a randomly selected ballot has been cast for the Democrat candidate, what is the probability that it was cast by a Republican voter? (b) What is the pdf of P (u) given that you observe one ballot and it was cast for a Republican? (c) What is the pdf of P (u) given a Republican wins the election? You can assume that a Republican win is equivalent to the condition that a Republican victory is more probable than a Democrat victory (i.e., from the condition in part (a)). 2.40. Consider the random variable Y (u) = g(X(u)) where the periodic function g(x) is shown in Fig. 5. If X(u) is uniform on [0, 4), determine and sketch the pdf of Y (u). Repeat for the case of X(u) is uniform on [0, 5 ). 2 2.41. Consider a game played in lecture. A student can choose to bet on a game or to abstain from betting. Students decide to bet with probability 0.8. If a student decides to bet, she wins with probability 0.6. When a student wins a bet, her point total is increased by 10; when she loses, it is decremented by 10. c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 20 g(x) 1 1 0 x 2 Figure 5: The periodic function considered in Problem 1.40. g(x) +4 +2 -3 -1 +1 +3 x -2 -4 Figure 6: An ideal model for a 2-bit analog to digital converter. (a) Let X(u) model the net change in the student's point total for one game as described above. Find the pmf of X(u) and determine the second moment description of X(u). (b) Suppose that one of these games is played each week for 10 weeks. Each week the game is played the same and the games are not related (i.e., independent trials). Let the event A be that a student has 5 wins, 2 losses, and 3 abstains in this 10 game series. Determine P (A). If A occurs, what is the net change in the student's point total? (c) Consider the case where 3 of these games have been played and let Y (u) model the net change to the student's point total. Find the pmf of Y (u) - rst specify the values {yk }k that Y (u) can take and then specify the corresponding probabilities. 2.42. A simple model for a 2-bit analog to digital converter (ADC) is given by the function in Fig. 6. This problem is concerned with the distortion introduced by this A/D conversion when the input is modeled as a zero mean Gaussian random variable. Specically, let X(u) be the ADC input, V (u) = g(X(u)) be the ADC output and Z(u) = X(u) V (u) be the approximation error. The input X(u) is modeled as Gaussian with zero mean and variance 2. (a) Determine the pdf of Z(u) c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 21 (b) Determine the following probabilities: Pr {|Z(u)| > 1} and Pr {0 < Z(u) < 1} 2.43. A class of 50 students has met for 9 lectures. Suppose that the instructor randomly selected 5 students at the beginning of each lecture and excused them from lecture. What is the probability that a specic student, say Jane, would have been excused from exactly 2 lectures thus far? 2.44. The IT Department at Ajax, Inc. is made up of 3 employees: Alice, Bob, and Sue. Ajax forms committees for various tasks by drawing from each department. In 2014, there will be 20 committees that require exactly one IT Department employee to participate. Which IT employee serves on a given committee is determined by drawing straws (i.e., randomly selecting between the 3). (a) Consider the event C that Alice, Bob, and Sue serve on 3, 7, 10 committees, respectively. Also, let the event D be that Alice, Bob, and Sue serve on 1, 0, 19 committees, respectively. Are these two events equally probable? Explain. (b) Find the probability that Alice serves on exactly m committees. What is the most probable number of committees Alice will serve on? (c) Repeat part (b) conditioned in the event E that Bob serves on exactly 10 committees. What is the most probable number of committees Alice will serve o given E? (d) Repeat part (b) conditioned in the event F that Bob serves on less than 3 committees. What is the most probable number of committees Alice will serve on given F ? 2.45. A study is being done by the legal department of an amusement park in order to assess potential liability. In particular, they are concerned with how well the roller-coaster's restraining system works for people of various heights. They have divided their potential customers into three categories, each of which they estimate is the same size: Adult Females, Adult Males, and Children. The height of each of these populations is modeled as Gaussian (in inches): Adult females have a mean height of 63 and a standard deviation of 4. Adult males have a mean height of 69 and a standard deviation of 6. Children have a mean height of 48 and a standard deviation of 12. The park is considering possible rules for prohibiting people from riding based on height. Each of the prohibitions below should be considered individually. (a) If the park prohibits people who are less than 36 inches tall, what is the probability that a randomly selected child will be prohibited from riding? (b) If the park prohibits people who are outside of the range (55, 71) inches tall, what is the probability that a randomly selected Adult Female will be prohibited from riding? (c) If the X(u) is the random variable modeling height, provide an expression for the pdf of X(u) and sketch this pdf. c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 22 (d) The most cautious of the lawyers suggests that only people between 60 and 71 inches in height should be allowed to ride. Find the probability that a randomly selected person will be prohibited from riding. (e) A person is selected at random and their height is 67 inches. Given this, what is the probability that the selected person is a Child, Adult Female, or Adult Male? 2.46. The game of \"Two Heads\" is played by ipping a coin until two heads have occurred. The probability of heads on a given ip is p and the ips are modeled as independent Bernoulli trials. Let X(u) be the random variable that models the number of ips required to obtain exactly 2 heads. (a) Specify the values {xk } that X(u) can take and determine the probability mass function (pmf) pX(u) (k) = Pr {X(u) = xk }. (b) If a fair coin is used, what is the probability that the two-heads condition is reach in 5 or fewer ips? Again, with a fair coin, what is the most probable value of X(u)? (c) Find the mean and variance of X(u) for general p. If a fair coin is used, what is the mean number of ips to reach the two-heads condition? 2.47. Let X(u) have pdf |x| e 2 where > 0 is a parameter. Determine and sketch the cdf for X(u). You have access to a random number generator that produces realizations of V (u) which is uniformly distributed on [0, 1] and you wish to generate Y (u) = g(V (u)) so that FY (u) (z) = FX(u) (z). Dene and sketch the function g(v). fX(u) (x) = 2.48. Consider the following compound random experiment. First, one of two coins is randomly selected from a bin containing one fair coin and one unfair coin. The fair coin comes up heads with probability 1/2 and the unfair coin comes up heads with probability p. Once a coin has been selected, it is ipped. If tails occurs, then a fair die is rolled once. If heads occurs, two fair dice are rolled. Let X(u) model the total number of dots on the die or dice. Determine and sketch the following: (a) Pr {X(u) = d|tails} (b) Pr {X(u) = d|heads} (c) Pr {X(u) = d} 2.49. A laboratory laser intensity measurement has been determined to be well-modeled by a Rayleigh random variable. Specically, this is modeled by the random variable R(u) with cdf given by 2 FR(u) (r) = (1 er /2 )U(r) It is desired to use this source of randomness to generate a Gaussian random variable. Specifically, a function g() is sought so that X(u) = g(R(u)) is standard Gaussian (i.e., zero mean and variance one). c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 23 (a) Determine Pr {R(u) < 1.2} and Pr {R(u) > 1.2} (b) Determine and sketch the desired function g(r) so that X(u) = g(R(u)) has a standard Gaussian random distribution. (c) If the value of r = 2.145 is measured what is g(r)? 2.50. You wish to learn about the probability of some event A. Initially you do not have any idea what this probability is so you model it as being uniformly distributed on [0, 1]. Specically, you model the probability of A occurring as a random variable P (u) with uniform distribution on [0, 1]. The goal is learn about this probability by accessing some data sets and determining in which sets A has occurred and in which sets A has not occurred. (a) Determine the mean and variance of P (u) and Pr {|P (u) 1/2| > 1/4} (b) Suppose that there are n observations available and A occurs in exactly k of these (k n) - let this event be denoted by Bn,k . Find an expression for the pdf of P (u) given Bn,k . Simplify as much as possible. (c) Consider the specic case of n = 6 and k = 4. Simplify and sketch the conditional pdf from the previous part. (d) Consider the same problem with n = 600 and k = 400. Use reasonable and accurate approximations to obtain a tractable expression for the result. 2.51. Consider scheduling a meeting where n people must attend. Assume that there are T possible meeting times to consider. Assume that each of the required attendees is available in each meeting time with probability p (i.e., otherwise they are already scheduled for another meeting). Also assume that availability across possible meeting times for a given person is independent and that all attendees have independent schedules. What is the probability that a meeting can be successfully scheduled? In other words, what is the probability that all required attendees are available for at least one meeting time? What is this probability when p = 3/5, n = 6, and T = 10? 2.52. Let V (u) be uniformly distributed on (0, 1). Conditioned on V (u) = v, X(u) is uniform on (0, v). Find fX(u) (x) and fV (u)|X(u) (v|x). 2.53. An investing club has n members. The club has decided to vote on whether or not to jointly invest in each of M dierent funds. For each fund, all n members vote. If no more than 1 member votes NO, then the club will invest in the fund. Investor preferences are modeled as independent from fund to fund with each investor voting to invest in a given fund with probability p. Investors are also assumed to vote independently of each other. What is the probability mass function for the number of funds in which the club invests? 2.54. Consider the following game: a coin is ipped until 3 heads and 2 tails are observed - denote 3 heads and 2 tails as a \"full house\". The probability of a heads for each ip is p. Let X(u) model the number of ips required to rst obtain a full house. (a) Determine the probability mass function for X(u) - i.e., pX(u) (k) = Pr {X(u) = zk } for each of the values zk from above. c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 24 fX(u) (x) 4 0 4 x Figure 7: The pdf of a random variable X(u). (b) If you were allowed to select the coin, what value of p would you select to maximize the probability that a full house is obtained in the rst 5 ips? What is the corresponding probability? (c) Consider this game using a fair coin. Observers can wager on the outcome of the game - e.g., bet that a full house is obtained in n or fewer coin ips. You are asked to design a strategy by a conservative bettor. The bettor wants to select n so that the probability that he wins is at least 0.75. Determine this value of n and the corresponding probability of a full house in n or fewer ips. 2.55. Consider the \"full house\" game from Prob. 2.54. Recall that one ips a coin, that lands on heads with probability p, until 3 heads and 2 tails are observed. Consider this game with a fair coin and with the following betting rules. A player bets $1 and she is returned: $1.50 is she ips a full house in 5 ips, $1.25 is she ips a full house in 6 ips, $1.00 is she ips a full house in 7 ips, $0 is she ips a full house in 8 or more ips. Let Z(u) model the net winnings for playing this game once - e.g., if she obtains a full house in 5 ips, her net winnings are $0.50 since she gets her dollar back, plus 50 cents. Find the probability of a positive net winnings and the mean and variance of Z(u). Does it make sense (nancially) to play this game, repeatedly, if you have a large amount of money to bet? 2.56. The continuous random variable X(u) has pdf as shown in Fig. 7. Determine the following: (a) mX 2 (b) X (c) Pr {X(u) > 0} (d) Pr {|X(u)| > 2} (e) E [X(u)]3 2.57. A coin is ipped n times and the number of heads is modeled by X(u). The probability that a heads occurs on each ip is p. (a) What is the probability mass function of X(u)? c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 25 (b) Let P (u) = X(u)/n be the random variable that models the fraction of ips that are heads. What is the mean and variance of P (u)? (c) Assuming that np(1 p) 1, determine approximate values for Pr |P (u) p| < . Assuming that the coin is fair and = 0.1, evaluate the above expression for the following values of n: 16, 100, 1024. 2.58. You have access to a random number generator that generates realizations of a random variable W (u) that is uniform on [, +]. It is desired to convert this to a random number generator that generates realizations of a Rayleigh random variable. A Rayleigh random variable X(u) has pdf given by fX(u) (x) = x x22 e 2 U(x) 2 where U(x) is the unit step function - i.e., the pdf is zero for negative values of x. Find a function h() such that h(W (u)) has the above Rayliegh distribution. 2.59. How much information is there in the roll of a fair die (in bits/roll)? How much info is there in a roll of a loaded die with: p(1) = 0.4, p(2) = 0.1, p(3) = 0.01, p(4) = 0.09, p(5) = 0.25, p(6) = 0.15? 2.60. What is the maximum rate for error free communication (capacity) of a binary symmetric channel with error probability = 0.4? How about of 0.2, 0.1, 0.01, 0.0001? 3 Pairs of Random Variables and Random Vectors 3.1. Sketch the region in the (X(u), Y (u))-plane corresponding to the following events: (a) {X(u) Y (u) 2} (b) {eX(u) < 6} (c) {max(X(u), Y (u)) < 6} (d) {min(X(u), Y (u)) < 6} (e) {|X(u) Y (u)| 2} (f) {|X(u)| > |Y (u)|} (g) {X(u)/Y (u) < 1} (h) {[X(u)]2 Y (u)} (i) {X(u)Y (u) 2} 3.2. Determine Pr {X(u) < Y (u)} and Pr {X(u) Y (u) 10} when fX(u)Y (u) (x, y) = 2e(x+2y) U(x)U(y). (17) 3.3. Determine the marginal pdf's of X(u) and Y (u) if fX(u)Y (u) (x, y) = xex(1+y) U(x)U(y). (18) c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 26 y 2 1 1 2 3 x Figure 8: The joint pdf considered in problem 3.5. 3.4. A fair coin is ipped 5 times. Let X(u) be the number of \"heads\" observed and let Y (u) = 4[X(u)]2 . Find the joint probability mass function for X(u) and Y (u). 3.5. Consider random variables X(u) and Y (u) with joint pdf function fX(u),Y (u) (x, y) = 1/4 in the shaded area and zero outside the shaded as shown in Fig. 8. (a) Find Pr {X(u) > Y (u)} (b) Find and sketch fX(u)|Y (u) (x|y). (c) Find Pr {1 < X(u) < 2|Y (u) = 0.5} and Pr {1 < X(u) < 2|Y (u) = 1.5} (d) Find Pr {1/4 < X(u) < 2|Y (u) = 0.1} and Pr {1 < X(u) < 2|Y (u) = 2.5} 3.6. The joint probability density function of X(u) and Y (u) (i.e., fX(u),Y (u) (x, y)) is equal to a constant K in {(x, y) : 0 x < 2, 0 y < 2}. Determine (a) K (b) The pdf of X(u) (c) Pr {0 Y (u) 1|X(u) = 0.5} 3.7. Determine the pdf of Z(u) = X(u)+Y (u) when X(u) and Y (u) are independent and uniformly distributed on [0, 1]. 3.8. The joint pdf of X(u) and Y (u) is 2 in the shaded region shown in Fig. 9 and 0 outside this region. (a) Find the marginal pdf's: fX(u) (x) and fY (u) (x) (b) Are X(u) and Y (u) statistically independent? (c) Let Z(u) = X(u) + Y (u), and nd the pdf of Z(u). Hint: This part is a little tedious; start by drawing the lines y = z x on the above plot for dierent values of z. Then perform the integration by considering the following cases separately: c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 27 y 1 0.5 1 0.5 x Figure 9: The joint pdf for problem 3.8. z z z z [0, 0.5) [0.5, 1) [1, 1.5) [1.5, 2) (d) Comparing your solution with problem 3.7, what is the conclusion? 3.9. USC maintenience stores light bulbs in either shed 1 or shed 2 with equal probability. Let S(u) be a random variable modeling which shed a bulb comes from (i.e. S(u) = i the bulb is from shed i). Let X(u) be the the time until the bulb burns out after installation. Bulbs from the dierent sheds fail with dierent probability: fX(u)|S(u) (x|1) = 2e2x U(x) fX(u)|S(u) (x|2) = 1/10 x [0, 10] 0 otherwise. Determine the following: (a) The mean of X(u) (b) Pr {S(u) = 1|X(u) = 5} (c) Pr {S(u) = 1|X(u) = 12} 3.10. The purpose of this problem is to verify some of the notation and results concerning jointlyGaussian random variables. X(u) and Y (u) are jointly-Gaussian if their joint-pdf is of the form exp fX(u)Y (u) (x, y) = 1 2(12 ) (xmX )2 2 X X )(ym 2 (xmX Y Y ) + 2X Y 1 2 (ymY )2 2 Y . c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 28 (a) Verify that the marginal densities are 2 fX(u) (x) = N (x; mX ; X ) 2 fY (u) (y) = N (y; mY ; Y ). Note that once you have veried one of the above equations the other follows by symmetry. (b) Verify that the above density can be written as fX(u)Y (u) (x, y) = N 2 (z; m; K) = 1 2 det(K) exp 1 (z m)t K1 (z m) , 2 where z= x y m= mX mY K= 2 X X Y X Y 2 Y and ()t denotes the transpose of a matrix/vector. What is the form of K when X(u) and Y (u) are independent? (c) Verify that the pdf of Y (u) conditioned on X(u) is fY (u)|X(u) (y|x) = N (y; mY |X (x); Y |X (x)2 ), (19) where Y (x mX ) X 2 = (1 2 )Y . mY |X (x) = mY + 2 Y |X (x)2 = Y |X Hint: For parts (a) and (c) you'll need to complete a square. Recall: z 2 2yz = (z y)2 y 2 . (20) 3.11. Let X1 (u) and X2 (u) be independent Gaussian random variables with zero mean variance 1. The random variables Y1 (u) and Y2 (u) are dened by Y1 (u) = 2X1 (u) + X2 (u) Y2 (u) = 3X1 (u) + 4X2 (u). Determine the following: c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 29 y 1 -1 1 x -1 Figure 10: The joint pdf considered in problem 3.12. (a) The marginal pdf of Y1 (u) (b) The joint pdf of Y1 (u) and Y2 (u). (c) fY1 (u)|Y2 (u) (y1 |y2 ) 3.12. The joint probability density function (pdf) of X(u) and Y (u), fXY (x, y), is nonzero, but not necessarily constant, only in the shaded region in Fig. 10. (a) Are X(u) and Y (u) independent? Is there enough information provided to answer this? (b) Are X(u) and Y (u) are uncorrelated? Is there enough information provided to answer this? (c) Is E |X(u)[Y (u)]3 | equal to, greater than, or less than 1? Is there enough information provided to answer this? 3.13. Listed below are random variables and the quantities which they are intended to model. In each case, determine whether the correlation coecient is positive, negative, or zero. (a) Dental Hygene: B = number of minutes per day a person spends brushing their teeth C = number of cavities a person has in a year (b) Performance Evaluation E = an employee's performance evaluation rating: 1 to 10, 10 being best S = employee salary c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 30 (c) Personal Numbers T = a person's telephone number W = a person's weight 3.14. For each of the joint-densities in problems 3.2 and 3.3, nd fY (u)|X(u) (y|x) and fX(u)|Y (u) (x|y). In each case also state whether the two random variables are independent or not. 3.15. Let X(u) and Y (u) be independent Exponential random variables, each with mean 1/, and let Z(u) = X(u) + Y (u). Determine the mean and the nth moment of Z(u). 3.16. Show that the sum of two independent Cauchy random variables is a Cauchy random variable. 3.17. The joint density of X(u) and Y (u) is 2 fX(u)Y (u) (x, y) = 2x2 exy ex U(x)U(y). (21) Determine the joint-pdf of the two random variables W (u) = X(u) Y (u) Z(u) = X(u)Y (u). 3.18. Let X(u) and Y (u) be jointly Gaussian random variables with means and variances: mX = 2, 2 2 X = 4, mY = 2, Y = 2, respectively. Also, let the correlation coeecent be X,Y = 0.5. Let Z(u) = 3X(u) + 4Y (u) 5 Determine the following the mean and variance of Z(u) and Pr {0 Z(u) 1}. 3.19. The normalized homework score (from 0 to 100%) can be modeled as a random variable H(u). The normalized test scores (from 0 to 100%) can be modeled as a random variable T (u). Based on past results, the second moment description of these random variables as mH = 48.1 H = 28.9 mT = 36.4 T = 17.6 E {(H(u) mH )(T (u) mT )} = 273 Assuming that these are good estimates (a) What is the correlation coecient between H(u) and T (u)? Explain the meaning. (b) Given your performance on the homework, predict your nal exam score. (c) How close is the exam score estimate to your actual midterm exam scores? Discuss the error - i.e., if your actual exam score is higher than the estimate, what does that say about your study habits? (d) What is the MSE of the exam score estimator? c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 31 3.20. The performance of a particular communication system is limited by interference from adjacent channels. The received signal is modeled as X(u) = S1 (u) + S2 (u), (22) where S1 (u) is the signal in the channel of interest and S2 (u) is the interfering signal from the adjacent channel. The constant represents an attenuation factor, so that || < 1. The signals on adjacent channels are not independent and are modeled as jointly-Gaussian random variables. Both S1 (u) and S2 (u) are zero mean and have variance 2 . The normalized correlation coecient for S1 (u) and S2 (u) is . Your task is design the best (Minimum Mean-Squared-Error) biased-linear estimator of S1 (u) based on observing X(u). (a) Determine the mean and variance of X(u) (b) Determine COV[X(u)S1 (u)] (c) Find the best linear estimate of S1 (u) based on observing X(u) - denoted by S1 (u) - and the corresponding minimum MSE. (d) While implementing the estimator you designed in part (b), a technician suggests that you can also obtain an estimate of the adjacent channel signal by intuition (the technician doesn't know any probability theory). The technician's estimate is 1 T S2 (u) = X(u) S1 (u) . (23) If you designed the best linear estimate of S2 (u) based on X(u) (denoted by S2 (u)), would you get the same estimate as the technician? 3.21. Let the joint pdf of X1 (u) and X2 (u) be fx(u) (x1 , x2 ) = 1 exp 2 3 1 (x1 1)2 (x1 1)(x2 2) + (x2 2)2 3 . (24) (a) Find mx and Kx . Hint: This is a joint-Gaussian pdf. (b) Any pair of jointly-Gaussian random variables can be transformed into two independent jointly-Gaussian random variables using a biased linear transformation. In this case show that Y1 (u) and Y2 (u) are independent Gaussians, when y(u) = Y1 (u) Y2 (u) 1 A= 2 = A(x(u) mx ) 1 1 1 1 . (c) Can you nd a matrix B so that z(u) = By(u) is a Gaussian random vector with Kz = I? (25) c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 32 3.22. Let X(u) and Z(u) be jointly-Gaussian random variables, each with mean zero and variance 1. Also, let be the correlation coecient between X(u) and Z(u). Given that X(u) is observed to be 0.91, what is the best MMSE estimate of Z(u)? Given this same realization of X(u), what is the best estimate of Z(u) based on a linear function of X(u)? 3.23. Consider random variables X(u) and Y (u) with joint pdf function fX(u),Y (u) (x, y) = 1/4 in the shaded area and zero outside the shaded as shown shown in Fig. 8. (a) Find the MMSE linear estimate of X(u) given Y = y. (b) Find the MMSE estimate of X(u) given Y = y. (c) Sketch the functions g(y) and h(y). 2 2 3.24. Let X(u) and Y (u) have variance 2 = X = Y . Find the variances and covariance for the random variables W (u) = (X(u) + Y (u))/ 2 and Z(u) = (X(u) Y (u))/ 2. Note, you can assume that X(u) and Y (u) have zero means. 3.25. Let X(u) and Y (u) be independent, zero mean Gaussian random variables, each with variance X(u)2 + Y (u)2 r . Discuss the relation to problem 2.8. 2 . Determine Pr 3.26. The joint pdf of X(u) and Y (u) is fX(u)Y (u) (x, y) = 1 exp 2 (x2 + y 2 ) 2 . (26) Determine the following: (a) Pr {|X(u)| 1, |Y (u)| 1} (b) fX(u) (x) 2 (c) mX and X 3.27. This problem addresses a method of converting two independent uniform random variable to two independent Gaussian random variables. Consider the independent uniformly distributed random variables X1 (u) and X2 (u) fX1 (u) (z) = fX2 (u) (z) = 1 z (0, 1) 0 otherwise. (27) The purpose of this problem is to demonstrate that the following are independent Gaussian random variables: Y1 (u) = Y2 (u) = 2 ln(X1 (u)) cos(2X2 (u)) 2 ln(X1 (u)) sin(2X2 (u)). (a) Determine the following: fX1 (u)X2 (u) (x1 , x2 ), E {Y1 (u)} and E {Y2 (u)}, and E {Y1 (u)Y2 (u)} c K.M. Chugg - October 2, 2015- Probability and Statistics Problem Set 33 j 3 2 1 0 0 1 2 3 i Figure 11: The region of nonzero joint pmf for problem 3.28. (b) Consider the random variable R(u) = mean of this random variable. 2 ln(X1 (u)). Determine the pdf fR(u) (r) and (c) Determine the joint density of Y1 (u) and Y2 (u), generated as described above (d) Answer the following questions: Are Are Are Are Are Are X1 (u) and X2 (u) uncorrelated? X1 (u) and X2 (u) orthogonal? Y1 (u) and Y2 (u) uncorrelated? Y1 (u) and Y2 (u) orthogonal? Y1 (u) and Y2 (u) independent? R(u) and X2 (u) independent? 3.28. The discrete random variables X(u) and Y (u) have joint pmf pX(u),Y (u) (i, j) that is nonzero only for the integer values of i and j as shown in Fig. 13. Furthermore, the pmf is equal to a constant C for these values of i and j. (a) Determine the constant C and Pr {Y (u) X(u) 1} (b) Determine and sketch the marginal pmfs of