Notes for Math 324, Part 17

Size: px
Start display at page:

Download "Notes for Math 324, Part 17"

Transcription

1 126 Notes for Math 324, Part 17

2 Chapter 17 Common discrete distributions 17.1 Binomial Consider an experiment consisting by a series of trials. The only possible outcomes of the trials are success and failure. The outcome of one trial is independent of the other trials. The probability of obtain a success is the same for all the trials. Let p be the probability of obtaining a success. A sequence of trials like this is called a Bernoulli sequence of trials. Let { 1 if the i th trial results in a success X i = 0 if the i th trial results in a failure The sequence of r.v. s X 1, X 2,... is called a Bernoulli sequence of r.v. s. Let S = n i=1 X i be the number of successes in the first n trials. Then, for each 0 k n, (17.1) P[S = k] = ( n k ) p k (1 p) n k. Here is the proof of (17.1). The probability of obtaining success, (k), success, failure, (n k), failure in precisely this order is p k (1 p) n k. However, the k successes and the n k failures can appear in any possible order. The number of ways to permutate k identical S s and n k identical F s is ( n k). Definition A r.v. S has discrete binomial distribution with parameters n and p if ( ) n P[S = k] = p k (1 p) n k, for 0 k n. k Here n is a positive integer and 0 p 1. Theorem Let S be a r.v. with a binomial distribution with parameters n and p. Then, E[S] = np and Var(S) = np(1 p). 127

3 128 CHAPTER 17. COMMON DISCRETE DISTRIBUTIONS Problem (# 40, Sample Test) A small commuter plane has 30 seats. The probability that any particular passenger will not show up for a flight is 0.10, independent of other passengers. The airline sells 32 tickets for the flight. Calculate the probability that more passengers show up for the flight than there are seats available. Answer: Solution: Let S be the number of passengers who show up. X has a binomial distribution with n = 32 and p = Then, P[more than 30 passengers show up] = P[S 31] = P[S = 31] + P[S = 32] = ( ) (0.90) 31 (0.1) + ( 32) (0.90) 32 = Problem (# 40, May 00) A company prices its hurricane insurance using the following assumptions: (i) In any calendar year, there can be at most one hurricane. (ii) In any calendar year, the probability of a hurricane is (iii) The number of hurricanes in any calendar year is independent of the number of hurricanes in any other calendar year. Using the company s assumptions, calculate the probability that there are fewer than 3 hurricanes in a -year period. Answer: 0.92 Solution: Let S be the number of hurricanes in years. X has a binomial distribution with parameters n = and p = We need to find P[S < 3] = P[S = 0] + P[S = 1] + P[S = 2] = ( ) 0 (0, 05) 0 (0.95) + ( ) 1 (0, 05) 1 (0.95) 19 + ( ) 2 (0, 05) 2 (0.95) 18 = Problem (# 37, May 01) A tour operator has a bus that can accommodate tourists. The operator knows that tourists may not show up, so he sells 21 tickets. The probability that an individual tourist will not show up is 0.02, independent of all other tourists. Each ticket costs 50, and is non-refundable if a tourist fails to show up. If a tourist shows up and a seat is not available, the tour operator has to pay 100 (ticket cost + 50 penalty) to the tourist. What is the expected revenue of the tour operator? Answer: 985 Solution: Let S be the number of tourists which show up. X has a binomial distribution with parameters n = 21 and p = The revenue of the tour operator is the r.v. Y = g(s), where { (21)(50) if S g(s) = (21)(50) 100 if S = 21 So, the expected revenue is E[Y ] = E[g(S)] = (21)(50)P[S ] + ((21)(50) 100)P [S = 21] = (21)(50)(1 P [S = 21]) + ((21)(50) 100)P [S = 21] = P [S = 21] = ( 21) (0.98) 21 = 985. Problem (# 27, November 01) A company establishes a fund of 1 from which it wants to pay an amount, C, to any of its employees who achieve a high performance level

4 17.1. BINOMIAL 129 during the coming year. Each employee has a 2% chance of achieving a high performance level during the coming year, independent of any other employee. Determine the maximum value of C for which the probability is less than 1% that the fund will be inadequate to cover all payments for high performance. Answer: 60 Solution: Let S be the number of employees who achieve a high performance level during the incoming year. S has a binomial distribution with parameters n = and p = The company pays a total of CS. We want to find the maximum value of C such that P[CS > 1] < This condition is equivalent to P[CS 1] We have that So, C = 60. P [S = 0] = ( ) 0 (.98) = P [S 1] = ( ) 0 (.98) + ( ) 1 (0.02)(.98) 19 = P [S 2] = ( ) 0 (.98) + ( ) 1 (0.02)(.98) 19 + ( ) 2 (0.02) 2 (0.98) 18 = Example Let X 1, X 2, X 3 be three independent continuous random variables each with density function { 2 x, if 0 < x < 2, f(x) = 0 else. What is the probability that exactly 2 of the 3 random variables exceeds 1? Solution: Let S be the number of r.v. s from X 1, X 2, X 3 which exceeds 1. S has a binomial distribution with n = 3 and p = P [X 1 1]. We have that p = P [X 1 1] = 2 1 Now, P[S = 2] = ( 3 2) (0.0858) 2 ( ) = ( ) ( ) 2x x x dx = = = Problem (# 23, May 01) A hospital receives 1/5 of its flu vaccine shipments from Company X and the remainder of its shipments from other companies. Each shipment contains a very large number of vaccine vials. For Company X s shipments, 10% of the vials are ineffective. For every other company, 2% of the vials are ineffective. The hospital tests 30 randomly selected vials from a shipment and finds that one vial is ineffective. What is the probability that this shipment came from Company X? Answer: 0.10 Solution: We apply Bayes theorem. Let B 1 be the event that the shipment came from Company X. Let B 2 be the event that the shipment did not come from Company X. Let A be the event that one vial in a sample of 30 is ineffective. By the Bayes theorem P[B 1 A] = P[B 1 ]P[A B 1 ] P[B 1 ]P[A B 1 ]+P[B 2 ]P[A B 2 ] We know that P[B 1 ] = 1 = 0.2 and P[B 5 2] = 4 = 0.8. We also have that 5 ( ) 30 P[A B 1 ] = P[Binom(n = 30, p = 0.1) = 1] = (0.1)(.9) 29 =

5 130 CHAPTER 17. COMMON DISCRETE DISTRIBUTIONS and So, P[A B 2 ] = P[Binom(n = 30, p = 0.02) = 1] = P[B 1 A] = (0.2)(0.1413) (0.2)(0.1413)+(0.8)(0.334) = ( ) 30 (0.02)(0.98) 29 = Geometric Consider a sequence of Bernoulli trials. Let Y be the number of the trial at which the first success occurs. Then, P[Y = 1] = P [success] = p, P[Y = 2] = P [failure, success] = (1 p)p, P[Y = 3] = P [failure, failure, success] = (1 p) 2 p, P[Y = 4] = P [failure, failure, failure, success] = (1 p) 3 p In general, for each k = 1, 2,..., Pr[Y = k] = (1 p) k 1 p. Definition A r.v. Y has a geometric distribution with parameter p, 0 p 1, if Pr[Y = k] = (1 p) k 1 p, k = 1, 2... Theorem Let Y be a geometric distribution with parameter p. Then, E[Y ] = 1 p and Var(Y ) = 1 p p 2. Proof. Taking the derivative to 1 = 1 x k=0 xk 1, 1 < x < 1, we get (1 x) 2 1 < x < 1. So, E[Y ] = k(1 p) k 1 p p = k=1 = k=1 kxk 1, (1 (1 p)) 2 = 1 p. Q.E.D. Example Let Y be the number of the throw of fair coin where the first head appears. Then, Y has a geometric distribution with p = 1. E[Y ] = 2 and Var(Y ) = 2. 2 Example Suppose that an ordinary six sided die is rolled repeatedly, and the outcome is noted in each roll. Assume that all the faces are equally probable. Let Y be the number of throw at which the first six occurs. Then, Y has a geometric distribution with p = 1 6. E[Y ] = 6 and Var(Y ) = 30. Let S n be the number of successes in the first n Bernoulli trials, we have that P[Y k] = P [S k 1 = 0] = (1 p) k 1. This is so, because {Y k} is equivalent to need k trials or more to obtain a success. {S k 1 = 0} equivalent to not obtain any success at all in the first k 1 trials. Example A certain basketball player hits a shot with probability 0.3. probability that he needs to throw 10 shots or more in order to get one shot? What is the

6 17.3. NEGATIVE BINOMIAL 131 Solution: We have to find P(Y 10) = P(S 9 = 0) = (0.7) 9. Let Y be a random variable Y with a geometric distribution, Then, for each k 1 For each k, n 1, So, P [Y k + n X k + 1] = P [Y k] = P [failure, (k 1), failure] = (1 p) k 1 P [Y k + n] P [Y k + 1] P [Y k + n 1 Y k] = P [Y n]. = (1 p)k+n 1 (1 p) k = (1 p) n 1 = P [Y n]. This formula says that if at the k 1 th trial we have not obtained a success, the probability that it will take an extra n or more trials to get a success is the same as the probability that it will take n or more trial to get a success for a Bernoulli sequences starting from the first trial. Having observed a roulette not giving red for a some trial does not change the probability of obtaining red in the next trials. Problem (# 7, Sample Test) As part of the underwriting process for insurance, each prospective policyholder is tested for high blood pressure. Let Y represent the number of tests completed when the first person with high blood pressure is found. The expected value of Y is Calculate the probability that the sixth person tested is the first one with high blood pressure. Answer: Solution: Let Y be the number of test at which the first person with high blood pressure is found. Then, Y has a geometric distribution with parameter p = P [a person has blood pressure]. E[Y ] = 1 p = So, p = We need to find P [Y = 6] = (1 0.08) 5 (0.08) = Negative binomial Consider a sequence of Bernoulli trials. Let Y r the number of the trail at which the r th sucess occurs, then ( ) k 1 P[Y r = k] = p r (1 p) k r, k = r, r + 1,.... r 1 This is so, because when {Y r = k} in the first k 1 trials, we obtain r 1 sucesses and in the k th trial we obtain another sucess. So, ( ) ( ) k 1 k 1 P[Y r = k] = p r 1 (1 p) (k 1) (r 1) p = p r (1 p) k r. r 1 r 1 A r.v. Y r having such as probability mass function is called a negative binomial distribution. Theorem E[Y r ] = r p and Var(Y r) = r(1 p) p 2

7 132 CHAPTER 17. COMMON DISCRETE DISTRIBUTIONS Let S n be the number of successes in the Bernoulli trials, we have that P[Y r k] = P (S k 1 r 1). This is so, because {Y r k} is equivalent to need k or more trials to obtain the r th success. This is the same as in the number of accumulated succeses until the k 1 th trial is less than r. This is the same as {S k 1 r 1}. Example Suppose that an fair six face die is rolled repeatedly, and the outcome is noted in each roll. What is the probability that the third 6 occurs on the seventh roll? Assume that all the faces are equally probable. Solution: Let Y 3 the number of the trial at which the third six occurs. Y 3 has a negative binomial distribution. We have that ( ) ( ) 3 ( ) P[Y 3 = 7] = Problem (# 11, November 01) A company takes out an insurance policy to cover accidents that occur at its manufacturing plant. The probability that one or more accidents will occur during any given month is 3. The number of accidents that occur in any given 5 month is independent of the number of accidents that occur in all other months. Calculate the probability that there will be at least four months in which no accidents occur before the fourth month in which at least one accident occurs. Answer: 0.29 Solution: Let S n be the number of months with accidents in the first n month. Let Y 4 be the number of the month when the fourth month with accidents occurs. We need to find P (Y 4 8). P [Y 4 8] = 1 P [Y 4 = 4] P [Y 4 = 5] P [Y 4 = 6] P [Y 4 = 7] 1 ( ) 3 3 (0.6) 4 (0.4) 0 ( ) 4 3 (0.6) 4 (0.4) 1 ( ) 5 3 (0.6) 4 (0.4) 2 ( 6 3) (0.6) 4 (0.4) 3 = Alternatively, using the relation between the negative binomial and the binomial, P[Y 4 8] = P [S 7 3] = P [S 7 = 0] + P [S 7 = 1] + P [S 7 = 2] + P [S 7 = 3] = ( ) 7 0 (0.6) 0 (.4) 7 + ( ) 7 1 (0.6) 1 (.4) 6 + ( ) 7 2 (0.6) 2 (.4) 5 + ( 7 3) (.06) 3 (.4) 4 = Poisson Definition A r.v. X has a Poisson distribution with parameter λ > 0, if for each integer k 0, In particular, λ λk P [X = k] = e k!. P [X = 0] = e λ, P[X = 1] = e λ λ, P [X = 2] = e λ λ2 2.

8 17.4. POISSON 133 Theorem Let X be a r.v. with a Poisson distribution with parameter λ > 0, then E[X] = λ and Var(X) = λ. Problem (# 24, May 00) An actuary has discovered that policyholders are three times as likely to file two claims as to file four claims. If the number of claims filed has a Poisson distribution, what is the variance of the number of claims filed? Answer: 2 Solution: Let X be the number of claims a poliholder files. X is a r.v. with a Poisson distribution with parameter λ > 0. We know that P [X = 2] = 3P [X = 4]. So, λ λ2 e 2 λ 2 = 4, and λ = 2. Hence, Var(X) = λ = 2. = 3e λ λ4 4! Problem (# 23, November 00) A company buys a policy to insure its revenue in the event of major snowstorms that shut down business. The policy pays nothing for the first such snowstorm of the year and 10, 000 for each one thereafter, until the end of the year. The number of major snowstorms per year that shut down business is assumed to have a Poisson distribution with mean 1.5. What is the expected amount paid to the company under this policy during a one-year period? Answer: 7,231 Solution: Let X be the number of number of major snowstorms per year that shut down business. X is a r.v. with a Poisson distribution with parameter λ = 1.5. The payment of the insurance company is Y = g(x), where { 0 if X 1 g(x) = 1000(X 1) if X 2 So, E[Y ] = E[g(X)] = E[1000(X 1)I(X 2)] = E[1000(X 1)] E[1000(X 1)I(X 1)] = 1000(E[X] 1) P(X = 0) = 1000(1.5 1) e 1.5 = Problem (# 19, November 01) A baseball team has scheduled its opening game for April 1. If it rains on April 1, the game is postponed and will be played on the next day that it does not rain. The team purchases insurance against rain. The policy will pay 1000 for each day, up to 2 days, that the opening game is postponed. The insurance company determines that the number of consecutive days of rain beginning on April 1 is a Poisson random variable with mean 0.6. What is the standard deviation of the amount the insurance company will have to pay? Answer: 699 Solution: Let X be the number of consecutive days of rain beginning on April 1. X is a r.v. with a Poisson distribution with parameter λ = 0.6. The payment made by the insurance company is Y = g(x), where 0 if X = 0 g(x) = 1000 if X = 1 00 if X 2

9 134 CHAPTER 17. COMMON DISCRETE DISTRIBUTIONS So, E[Y ] = E[g(X)] = 1000P(X = 1) + 00P(X 2) = 1000P(X = 1) + 00(1 P(X = 0) P(X = 1)) = 1000 e 0.6 (0.6) + 00(1 e 0.6 e 0.6 (0.6)) = E[Y 2 ] = E[(g(X)) 2 ] = P(X = 1) P(X 2) = P(X = 1) (1 P(X = 0) P(X = 1)) = e 0.6 (0.6) (1 e 0.6 e 0.6 (0.6)) = So, the standard deviation of the amount the insurance company will have to pay is σ = E[Y 2 ] (E[Y ]) 2 = ( ) 2 = Hypergeometric Consider an urn with N white balls and M white balls, k are withdrawn from the urn without replacement. Let X be the number of white balls selected, then ( N )( M ) i k i P[X = i] = ) ( N+M k for 0 i N and 0 k i M. The distribution of this r.v. is called hypergeometric. Definition A r.v. X has a hypergeometric distribution with parameters N, M and k if ( N )( M ) i k i P[X = i] = ) for max(k M, 0) i min(n, k) ( N+M k Theorem Let X be a r.v. hypergeometric distribution with parameters N, M and k, then E[X] = kn. N+M 17.6 Problems 1. An analysis of auto accidents shows that one in four accidents results in an insurance claim. In a series of independent accidents, find the probability that the first accident resulting in an insurance claim is one of the first 3 accidents. 2. If X has a Poisson distribution of that 3P [X = 1] = P [X = 2]. Find P [X = 4]. 3. A manufacturer of soft drink bottles turns out defective with probability Assume that every bottle turn out defective independently of the rest of the bottles produced. Find the probability that there are exactly 4 defective bottles among the next 10 bottles produced. 4. Thirty percent of the applicants for a position have advanced training in computer programming. Three jobs requiring advanced training are open. Find the probability that the third qualified applicant is found on the fifth interview, supposing the applicants are interviewed sequentially and at random.

10 17.6. PROBLEMS The lifetime in hours of a certain kind of radio tube is a random variable having a probability density function given by { 100 if x > 100, x f(x) = 2 0 else, What is the probability that exactly 2 of 5 such tubes in radio set will have to be replaced within the first 150 hours of operation? 6. A commuter s drive to work includes 7 stoplights. Assume that probability that a light is red when the commuter reaches it is 0., and that the lights are far enough apart to operate independently. Let X be the number of red lights the commuter stops for. Find the probability that the commuter has to stop for 2 or more lights. Find the expected value of X. 7. A system of 50 components functions if 45 or more of their components functions properly. Find the probability that the system operates if the probability that a component operates properly is 0.85, and different components operate or not independently of the rest. 8. A box contains 35 gems, of which 10 are real diamonds and 25 are fake diamonds. Gems are randomly taking out of the box, one at a time without replacement. What is the probability that exactly 2 fakes are selected before the second real diamond is selected? 9. A ball is drawn from an urn containing 4 white and 2 black balls. After the ball is drawn, it is then replaced and another ball is drawn. This goes indefinitely. What is the probability that, of the first 8 balls drawn, exactly 3 are white? 10. A certain basketball player hits a shot with probability 0.3. What is the probability that he needs to throw 10 or more shots in order to get 4 hits? 11. In an initial screening of job applicants, a recruiter accepts, on the average, one-third of all applicants for further consideration. In reviewing a collection of job applications (independent of one another), find the probability that the first application acceptable for further consideration is one of the first three applications reviewed. 12. On a six question multiple choice test there are five possible answers of which one is correct and four incorrect. If a student guesses randomly and independently find the probability that he gets at least 2 questions right? 13. Suppose that an ordinary six sided fair die is rolled repeatedly, and the outcome is noted in each roll. What is the probability that the third 6 occurs on the seventh roll? 14. What is more probable to get 2 or more sixes in 6 throws of a fair die or to get or more sixes in 60 throws of a fair die? Find each of these probabilities. 15. If the probability of hitting a target is 1 5, and then ten shoots are fired independently, what is the probability of the target being hit at least twice?

11 136 CHAPTER 17. COMMON DISCRETE DISTRIBUTIONS 16. A basketball player shoots 10 shots and the probability of hitting is 0.4 in on each shot. What is the probability of hitting exactly eight shots? 17. If X is a binomial random variable with expected value 6 and variance 2.4, find P (X = 5). 18. A fair die is tossed until a 2 is obtained. If X is the number of trials required to obtain the first 2, what is the smallest value of x for which P (X x) 1/ A certain hunter hits a duck with probability 0.3. What is the probability that he needs to throw 10 or more shots in order to get 2 or more ducks?. A ball is drawn from an urn containing 4 white and 2 black balls. After the ball is drawn, it is then replaced and another ball is drawn. This goes indefinitely. What is the probability that, of the first 8 balls drawn, exactly 3 are white? 21. The lifetime in hours of a certain kind of radio tube is a random variable having a probability density function given by { 100 for x > 100 x f(x) = 2 0 else A radio set has 50 of such tubes. What is the expected number of tubes in the radio set which will be still working after 150 hours of operation? 22. A man pays $1 a throw to try to win a $3 doll. His probability of winning on each throw is 0.1. What is the probability that more than three throws will be required to win the doll? 23. If a student selects true or false at random on an examination, assuming independence among the answers, determine the probability that at most three questions must be answered at random to obtain the first correct answer. 24. A deck of 52 cards is dealt out. What is the probability that the second ace occurs on the fourteenth card? 25. An experiment consists of tossing a fair die until a 6 occurs four times. What is the probability that the process ends after exactly ten tosses with a 6 occurring on the ninth and tenth tosses? 26. Find the probability that the fifth head is observed on the tenth independent flip of an unbiased coin. 27. A fair coin is flipped 4 times. What is the probability that the fourth flip is a head, given that each of the first flips resulted in heads? 28. Suppose that in 4 child families, each child is equally likely to be a boy or a girl, independently of the others. Which would then be more common, 4 child families with 2 boys and 2 girls, or 4 child families with different numbers of boys and girls?

12 17.6. PROBLEMS Suppose that an ordinary six sided die is rolled repeatedly, and the outcome is noted in each roll. What is the probability that the third 6 occurs on the seventh roll? Assume that all the faces are equally probable.

1. If X has density. cx 3 e x ), 0 x < 0, otherwise. Find the value of c that makes f a probability density. f(x) =

1. If X has density. cx 3 e x ), 0 x < 0, otherwise. Find the value of c that makes f a probability density. f(x) = 1. If X has density f(x) = { cx 3 e x ), 0 x < 0, otherwise. Find the value of c that makes f a probability density. 2. Let X have density f(x) = { xe x, 0 < x < 0, otherwise. (a) Find P (X > 2). (b) Find

More information

Discrete Distributions

Discrete Distributions A simplest example of random experiment is a coin-tossing, formally called Bernoulli trial. It happens to be the case that many useful distributions are built upon this simplest form of experiment, whose

More information

Notes for Math 324, Part 20

Notes for Math 324, Part 20 7 Notes for Math 34, Part Chapter Conditional epectations, variances, etc.. Conditional probability Given two events, the conditional probability of A given B is defined by P[A B] = P[A B]. P[B] P[A B]

More information

Math st Homework. First part of Chapter 2. Due Friday, September 17, 1999.

Math st Homework. First part of Chapter 2. Due Friday, September 17, 1999. Math 447. 1st Homework. First part of Chapter 2. Due Friday, September 17, 1999. 1. How many different seven place license plates are possible if the first 3 places are to be occupied by letters and the

More information

Notes for Math 324, Part 19

Notes for Math 324, Part 19 48 Notes for Math 324, Part 9 Chapter 9 Multivariate distributions, covariance Often, we need to consider several random variables at the same time. We have a sample space S and r.v. s X, Y,..., which

More information

Math 447. Introduction to Probability and Statistics I. Fall 1998.

Math 447. Introduction to Probability and Statistics I. Fall 1998. Math 447. Introduction to Probability and Statistics I. Fall 1998. Schedule: M. W. F.: 08:00-09:30 am. SW 323 Textbook: Introduction to Mathematical Statistics by R. V. Hogg and A. T. Craig, 1995, Fifth

More information

Some Special Discrete Distributions

Some Special Discrete Distributions Mathematics Department De La Salle University Manila February 6, 2017 Some Discrete Distributions Often, the observations generated by different statistical experiments have the same general type of behaviour.

More information

Exercises in Probability Theory Paul Jung MA 485/585-1C Fall 2015 based on material of Nikolai Chernov

Exercises in Probability Theory Paul Jung MA 485/585-1C Fall 2015 based on material of Nikolai Chernov Exercises in Probability Theory Paul Jung MA 485/585-1C Fall 2015 based on material of Nikolai Chernov Many of the exercises are taken from two books: R. Durrett, The Essentials of Probability, Duxbury

More information

Find the value of n in order for the player to get an expected return of 9 counters per roll.

Find the value of n in order for the player to get an expected return of 9 counters per roll. . A biased die with four faces is used in a game. A player pays 0 counters to roll the die. The table below shows the possible scores on the die, the probability of each score and the number of counters

More information

MATH 250 / SPRING 2011 SAMPLE QUESTIONS / SET 3

MATH 250 / SPRING 2011 SAMPLE QUESTIONS / SET 3 MATH 250 / SPRING 2011 SAMPLE QUESTIONS / SET 3 1. A four engine plane can fly if at least two engines work. a) If the engines operate independently and each malfunctions with probability q, what is the

More information

HW on Ch Let X be a discrete random variable with V (X) = 8.6, then V (3X+5.6) is. V (3X + 5.6) = 3 2 V (X) = 9(8.6) = 77.4.

HW on Ch Let X be a discrete random variable with V (X) = 8.6, then V (3X+5.6) is. V (3X + 5.6) = 3 2 V (X) = 9(8.6) = 77.4. HW on Ch 3 Name: Questions:. Let X be a discrete random variable with V (X) = 8.6, then V (3X+5.6) is. V (3X + 5.6) = 3 2 V (X) = 9(8.6) = 77.4. 2. Let X be a discrete random variable with E(X 2 ) = 9.75

More information

Tutorial 3 - Discrete Probability Distributions

Tutorial 3 - Discrete Probability Distributions Tutorial 3 - Discrete Probability Distributions 1. If X ~ Bin(6, ), find (a) P(X = 4) (b) P(X 2) 2. If X ~ Bin(8, 0.4), find (a) P(X = 2) (b) P(X = 0) (c)p(x > 6) 3. The probability that a pen drawn at

More information

Probabilistic models

Probabilistic models Probabilistic models Kolmogorov (Andrei Nikolaevich, 1903 1987) put forward an axiomatic system for probability theory. Foundations of the Calculus of Probabilities, published in 1933, immediately became

More information

Discrete Random Variable Practice

Discrete Random Variable Practice IB Math High Level Year Discrete Probability Distributions - MarkScheme Discrete Random Variable Practice. A biased die with four faces is used in a game. A player pays 0 counters to roll the die. The

More information

Chapter 17 Probability Models

Chapter 17 Probability Models Chapter 17 Probability Models 241 Chapter 17 Probability Models 1 Bernoulli a) These are not Bernoulli trials The possible outcomes are 1, 2, 3, 4, 5, and There are more than two possible outcomes b) These

More information

DISCRETE VARIABLE PROBLEMS ONLY

DISCRETE VARIABLE PROBLEMS ONLY DISCRETE VARIABLE PROBLEMS ONLY. A biased die with four faces is used in a game. A player pays 0 counters to roll the die. The table below shows the possible scores on the die, the probability of each

More information

TOPIC 12 PROBABILITY SCHEMATIC DIAGRAM

TOPIC 12 PROBABILITY SCHEMATIC DIAGRAM TOPIC 12 PROBABILITY SCHEMATIC DIAGRAM Topic Concepts Degree of Importance References NCERT Book Vol. II Probability (i) Conditional Probability *** Article 1.2 and 1.2.1 Solved Examples 1 to 6 Q. Nos

More information

Probabilistic models

Probabilistic models Kolmogorov (Andrei Nikolaevich, 1903 1987) put forward an axiomatic system for probability theory. Foundations of the Calculus of Probabilities, published in 1933, immediately became the definitive formulation

More information

What is Probability? Probability. Sample Spaces and Events. Simple Event

What is Probability? Probability. Sample Spaces and Events. Simple Event What is Probability? Probability Peter Lo Probability is the numerical measure of likelihood that the event will occur. Simple Event Joint Event Compound Event Lies between 0 & 1 Sum of events is 1 1.5

More information

Course 1 Solutions November 2001 Exams

Course 1 Solutions November 2001 Exams Course Solutions November Exams . A For i =,, let R = event that a red ball is drawn form urn i i B = event that a blue ball is drawn from urn i. i Then if x is the number of blue balls in urn, ( R R)

More information

Expected Value 7/7/2006

Expected Value 7/7/2006 Expected Value 7/7/2006 Definition Let X be a numerically-valued discrete random variable with sample space Ω and distribution function m(x). The expected value E(X) is defined by E(X) = x Ω x m(x), provided

More information

Notes slides from before lecture. CSE 21, Winter 2017, Section A00. Lecture 16 Notes. Class URL:

Notes slides from before lecture. CSE 21, Winter 2017, Section A00. Lecture 16 Notes. Class URL: Notes slides from before lecture CSE 21, Winter 2017, Section A00 Lecture 16 Notes Class URL: http://vlsicad.ucsd.edu/courses/cse21-w17/ Notes slides from before lecture Notes March 8 (1) This week: Days

More information

Chapter 8 Sequences, Series, and Probability

Chapter 8 Sequences, Series, and Probability Chapter 8 Sequences, Series, and Probability Overview 8.1 Sequences and Series 8.2 Arithmetic Sequences and Partial Sums 8.3 Geometric Sequences and Partial Sums 8.5 The Binomial Theorem 8.6 Counting Principles

More information

November 2000 Course 1. Society of Actuaries/Casualty Actuarial Society

November 2000 Course 1. Society of Actuaries/Casualty Actuarial Society November 2000 Course 1 Society of Actuaries/Casualty Actuarial Society 1. A recent study indicates that the annual cost of maintaining and repairing a car in a town in Ontario averages 200 with a variance

More information

Question Points Score Total: 137

Question Points Score Total: 137 Math 447 Test 1 SOLUTIONS Fall 2015 No books, no notes, only SOA-approved calculators. true/false or fill-in-the-blank question. You must show work, unless the question is a Name: Section: Question Points

More information

Homework 4 Solution, due July 23

Homework 4 Solution, due July 23 Homework 4 Solution, due July 23 Random Variables Problem 1. Let X be the random number on a die: from 1 to. (i) What is the distribution of X? (ii) Calculate EX. (iii) Calculate EX 2. (iv) Calculate Var

More information

Statistics for Economists. Lectures 3 & 4

Statistics for Economists. Lectures 3 & 4 Statistics for Economists Lectures 3 & 4 Asrat Temesgen Stockholm University 1 CHAPTER 2- Discrete Distributions 2.1. Random variables of the Discrete Type Definition 2.1.1: Given a random experiment with

More information

Week 6, 9/24/12-9/28/12, Notes: Bernoulli, Binomial, Hypergeometric, and Poisson Random Variables

Week 6, 9/24/12-9/28/12, Notes: Bernoulli, Binomial, Hypergeometric, and Poisson Random Variables Week 6, 9/24/12-9/28/12, Notes: Bernoulli, Binomial, Hypergeometric, and Poisson Random Variables 1 Monday 9/24/12 on Bernoulli and Binomial R.V.s We are now discussing discrete random variables that have

More information

ELEG 3143 Probability & Stochastic Process Ch. 2 Discrete Random Variables

ELEG 3143 Probability & Stochastic Process Ch. 2 Discrete Random Variables Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 2 Discrete Random Variables Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Random Variable Discrete Random

More information

An-Najah National University Faculty of Engineering Industrial Engineering Department. Course : Quantitative Methods (65211)

An-Najah National University Faculty of Engineering Industrial Engineering Department. Course : Quantitative Methods (65211) An-Najah National University Faculty of Engineering Industrial Engineering Department Course : Quantitative Methods (65211) Instructor: Eng. Tamer Haddad 2 nd Semester 2009/2010 Chapter 3 Discrete Random

More information

Chapter 16 Probability Models

Chapter 16 Probability Models Chapter 16 Probability Models Chapter 16 Probability Models 301 1. Bernoulli. a) These are not Bernoulli trials. The possible outcomes are 1, 2, 3, 4, 5, and 6. There are more than two possible outcomes.

More information

Week 04 Discussion. a) What is the probability that of those selected for the in-depth interview 4 liked the new flavor and 1 did not?

Week 04 Discussion. a) What is the probability that of those selected for the in-depth interview 4 liked the new flavor and 1 did not? STAT Wee Discussion Fall 7. A new flavor of toothpaste has been developed. It was tested by a group of people. Nine of the group said they lied the new flavor, and the remaining 6 indicated they did not.

More information

Conditional Probability

Conditional Probability Conditional Probability Idea have performed a chance experiment but don t know the outcome (ω), but have some partial information (event A) about ω. Question: given this partial information what s the

More information

Probability and Statistics Notes

Probability and Statistics Notes Probability and Statistics Notes Chapter One Jesse Crawford Department of Mathematics Tarleton State University (Tarleton State University) Chapter One Notes 1 / 71 Outline 1 A Sketch of Probability and

More information

Topic 3: The Expectation of a Random Variable

Topic 3: The Expectation of a Random Variable Topic 3: The Expectation of a Random Variable Course 003, 2017 Page 0 Expectation of a discrete random variable Definition (Expectation of a discrete r.v.): The expected value (also called the expectation

More information

Chapter 2: Discrete Distributions. 2.1 Random Variables of the Discrete Type

Chapter 2: Discrete Distributions. 2.1 Random Variables of the Discrete Type Chapter 2: Discrete Distributions 2.1 Random Variables of the Discrete Type 2.2 Mathematical Expectation 2.3 Special Mathematical Expectations 2.4 Binomial Distribution 2.5 Negative Binomial Distribution

More information

Geometric Distribution The characteristics of a geometric experiment are: 1. There are one or more Bernoulli trials with all failures except the last

Geometric Distribution The characteristics of a geometric experiment are: 1. There are one or more Bernoulli trials with all failures except the last Geometric Distribution The characteristics of a geometric experiment are: 1. There are one or more Bernoulli trials with all failures except the last one, which is a success. In other words, you keep repeating

More information

Section 2.4 Bernoulli Trials

Section 2.4 Bernoulli Trials Section 2.4 Bernoulli Trials A bernoulli trial is a repeated experiment with the following properties: 1. There are two outcomes of each trial: success and failure. 2. The probability of success in each

More information

Definition: A random variable X is a real valued function that maps a sample space S into the space of real numbers R. X : S R

Definition: A random variable X is a real valued function that maps a sample space S into the space of real numbers R. X : S R Random Variables Definition: A random variable X is a real valued function that maps a sample space S into the space of real numbers R. X : S R As such, a random variable summarizes the outcome of an experiment

More information

HYPERGEOMETRIC and NEGATIVE HYPERGEOMETIC DISTRIBUTIONS

HYPERGEOMETRIC and NEGATIVE HYPERGEOMETIC DISTRIBUTIONS HYPERGEOMETRIC and NEGATIVE HYPERGEOMETIC DISTRIBUTIONS A The Hypergeometric Situation: Sampling without Replacement In the section on Bernoulli trials [top of page 3 of those notes], it was indicated

More information

Discussion 03 Solutions

Discussion 03 Solutions STAT Discussion Solutions Spring 8. A new flavor of toothpaste has been developed. It was tested by a group of people. Nine of the group said they liked the new flavor, and the remaining indicated they

More information

Math 151. Rumbos Fall Solutions to Review Problems for Final Exam

Math 151. Rumbos Fall Solutions to Review Problems for Final Exam Math 5. Rumbos Fall 23 Solutions to Review Problems for Final Exam. Three cards are in a bag. One card is red on both sides. Another card is white on both sides. The third card in red on one side and white

More information

BINOMIAL DISTRIBUTION

BINOMIAL DISTRIBUTION BINOMIAL DISTRIBUTION The binomial distribution is a particular type of discrete pmf. It describes random variables which satisfy the following conditions: 1 You perform n identical experiments (called

More information

STAT/MA 416 Answers Homework 4 September 27, 2007 Solutions by Mark Daniel Ward PROBLEMS

STAT/MA 416 Answers Homework 4 September 27, 2007 Solutions by Mark Daniel Ward PROBLEMS STAT/MA 416 Answers Homework 4 September 27, 2007 Solutions by Mark Daniel Ward PROBLEMS 2. We ust examine the 36 possible products of two dice. We see that 1/36 for i = 1, 9, 16, 25, 36 2/36 for i = 2,

More information

Chapter (4) Discrete Probability Distributions Examples

Chapter (4) Discrete Probability Distributions Examples Chapter (4) Discrete Probability Distributions Examples Example () Two balanced dice are rolled. Let X be the sum of the two dice. Obtain the probability distribution of X. Solution When the two balanced

More information

Part (A): Review of Probability [Statistics I revision]

Part (A): Review of Probability [Statistics I revision] Part (A): Review of Probability [Statistics I revision] 1 Definition of Probability 1.1 Experiment An experiment is any procedure whose outcome is uncertain ffl toss a coin ffl throw a die ffl buy a lottery

More information

STA 2023 EXAM-2 Practice Problems From Chapters 4, 5, & Partly 6. With SOLUTIONS

STA 2023 EXAM-2 Practice Problems From Chapters 4, 5, & Partly 6. With SOLUTIONS STA 2023 EXAM-2 Practice Problems From Chapters 4, 5, & Partly 6 With SOLUTIONS Mudunuru Venkateswara Rao, Ph.D. STA 2023 Fall 2016 Venkat Mu ALL THE CONTENT IN THESE SOLUTIONS PRESENTED IN BLUE AND BLACK

More information

Math 493 Final Exam December 01

Math 493 Final Exam December 01 Math 493 Final Exam December 01 NAME: ID NUMBER: Return your blue book to my office or the Math Department office by Noon on Tuesday 11 th. On all parts after the first show enough work in your exam booklet

More information

Chapter 3 Discrete Random Variables

Chapter 3 Discrete Random Variables MICHIGAN STATE UNIVERSITY STT 351 SECTION 2 FALL 2008 LECTURE NOTES Chapter 3 Discrete Random Variables Nao Mimoto Contents 1 Random Variables 2 2 Probability Distributions for Discrete Variables 3 3 Expected

More information

Math 218 Supplemental Instruction Spring 2008 Final Review Part A

Math 218 Supplemental Instruction Spring 2008 Final Review Part A Spring 2008 Final Review Part A SI leaders: Mario Panak, Jackie Hu, Christina Tasooji Chapters 3, 4, and 5 Topics Covered: General probability (probability laws, conditional, joint probabilities, independence)

More information

107 Exercises in Probability Theory

107 Exercises in Probability Theory UNIVERSITY OF KENT Institute of Mathematics, Statistics and Actuarial Science Module MA304 DISCRETE MATHEMATICS AND PROBABILITY 107 Exercises in Probability Theory 1 2 1. Suppose that the sample space

More information

2014 SM4 Revision Questions Distributions

2014 SM4 Revision Questions Distributions 2014 SM4 Revision Questions Distributions Normal Q1. Professor Halen has 184 students in his college mathematics class. The scores on the semester exam are normally distributed with a mean of 72.3 and

More information

STA 2023 EXAM-2 Practice Problems. Ven Mudunuru. From Chapters 4, 5, & Partly 6. With SOLUTIONS

STA 2023 EXAM-2 Practice Problems. Ven Mudunuru. From Chapters 4, 5, & Partly 6. With SOLUTIONS STA 2023 EXAM-2 Practice Problems From Chapters 4, 5, & Partly 6 With SOLUTIONS Mudunuru, Venkateswara Rao STA 2023 Spring 2016 1 1. A committee of 5 persons is to be formed from 6 men and 4 women. What

More information

Each trial has only two possible outcomes success and failure. The possible outcomes are exactly the same for each trial.

Each trial has only two possible outcomes success and failure. The possible outcomes are exactly the same for each trial. Section 8.6: Bernoulli Experiments and Binomial Distribution We have already learned how to solve problems such as if a person randomly guesses the answers to 10 multiple choice questions, what is the

More information

Analysis of Engineering and Scientific Data. Semester

Analysis of Engineering and Scientific Data. Semester Analysis of Engineering and Scientific Data Semester 1 2019 Sabrina Streipert s.streipert@uq.edu.au Example: Draw a random number from the interval of real numbers [1, 3]. Let X represent the number. Each

More information

Chapter 2 Random Variables

Chapter 2 Random Variables Stochastic Processes Chapter 2 Random Variables Prof. Jernan Juang Dept. of Engineering Science National Cheng Kung University Prof. Chun-Hung Liu Dept. of Electrical and Computer Eng. National Chiao Tung

More information

Introduction and Overview STAT 421, SP Course Instructor

Introduction and Overview STAT 421, SP Course Instructor Introduction and Overview STAT 421, SP 212 Prof. Prem K. Goel Mon, Wed, Fri 3:3PM 4:48PM Postle Hall 118 Course Instructor Prof. Goel, Prem E mail: goel.1@osu.edu Office: CH 24C (Cockins Hall) Phone: 614

More information

STAT 414: Introduction to Probability Theory

STAT 414: Introduction to Probability Theory STAT 414: Introduction to Probability Theory Spring 2016; Homework Assignments Latest updated on April 29, 2016 HW1 (Due on Jan. 21) Chapter 1 Problems 1, 8, 9, 10, 11, 18, 19, 26, 28, 30 Theoretical Exercises

More information

Solutionbank S1 Edexcel AS and A Level Modular Mathematics

Solutionbank S1 Edexcel AS and A Level Modular Mathematics Heinemann Solutionbank: Statistics S Page of Solutionbank S Exercise A, Question Write down whether or not each of the following is a discrete random variable. Give a reason for your answer. a The average

More information

Example. What is the sample space for flipping a fair coin? Rolling a 6-sided die? Find the event E where E = {x x has exactly one head}

Example. What is the sample space for flipping a fair coin? Rolling a 6-sided die? Find the event E where E = {x x has exactly one head} Chapter 7 Notes 1 (c) Epstein, 2013 CHAPTER 7: PROBABILITY 7.1: Experiments, Sample Spaces and Events Chapter 7 Notes 2 (c) Epstein, 2013 What is the sample space for flipping a fair coin three times?

More information

STAT2201. Analysis of Engineering & Scientific Data. Unit 3

STAT2201. Analysis of Engineering & Scientific Data. Unit 3 STAT2201 Analysis of Engineering & Scientific Data Unit 3 Slava Vaisman The University of Queensland School of Mathematics and Physics What we learned in Unit 2 (1) We defined a sample space of a random

More information

Special Mathematics Discrete random variables

Special Mathematics Discrete random variables Special Mathematics Discrete random variables April 208 ii Expose yourself to as much randomness as possible. Ben Casnocha 6 Discrete random variables Texas Holdem Poker: In Hold em Poker players make

More information

Relationship between probability set function and random variable - 2 -

Relationship between probability set function and random variable - 2 - 2.0 Random Variables A rat is selected at random from a cage and its sex is determined. The set of possible outcomes is female and male. Thus outcome space is S = {female, male} = {F, M}. If we let X be

More information

Intermediate Math Circles November 8, 2017 Probability II

Intermediate Math Circles November 8, 2017 Probability II Intersection of Events and Independence Consider two groups of pairs of events Intermediate Math Circles November 8, 017 Probability II Group 1 (Dependent Events) A = {a sales associate has training} B

More information

Statistics for Managers Using Microsoft Excel/SPSS Chapter 4 Basic Probability And Discrete Probability Distributions

Statistics for Managers Using Microsoft Excel/SPSS Chapter 4 Basic Probability And Discrete Probability Distributions Statistics for Managers Using Microsoft Excel/SPSS Chapter 4 Basic Probability And Discrete Probability Distributions 1999 Prentice-Hall, Inc. Chap. 4-1 Chapter Topics Basic Probability Concepts: Sample

More information

Dept. of Linguistics, Indiana University Fall 2015

Dept. of Linguistics, Indiana University Fall 2015 L645 Dept. of Linguistics, Indiana University Fall 2015 1 / 34 To start out the course, we need to know something about statistics and This is only an introduction; for a fuller understanding, you would

More information

COVENANT UNIVERSITY NIGERIA TUTORIAL KIT OMEGA SEMESTER PROGRAMME: ECONOMICS

COVENANT UNIVERSITY NIGERIA TUTORIAL KIT OMEGA SEMESTER PROGRAMME: ECONOMICS COVENANT UNIVERSITY NIGERIA TUTORIAL KIT OMEGA SEMESTER PROGRAMME: ECONOMICS COURSE: CBS 221 DISCLAIMER The contents of this document are intended for practice and leaning purposes at the undergraduate

More information

Edexcel past paper questions

Edexcel past paper questions Edexcel past paper questions Statistics 1 Discrete Random Variables Past examination questions Discrete Random variables Page 1 Discrete random variables Discrete Random variables Page 2 Discrete Random

More information

4. Suppose that we roll two die and let X be equal to the maximum of the two rolls. Find P (X {1, 3, 5}) and draw the PMF for X.

4. Suppose that we roll two die and let X be equal to the maximum of the two rolls. Find P (X {1, 3, 5}) and draw the PMF for X. Math 10B with Professor Stankova Worksheet, Midterm #2; Wednesday, 3/21/2018 GSI name: Roy Zhao 1 Problems 1.1 Bayes Theorem 1. Suppose a test is 99% accurate and 1% of people have a disease. What is the

More information

STAT 430/510 Probability Lecture 7: Random Variable and Expectation

STAT 430/510 Probability Lecture 7: Random Variable and Expectation STAT 430/510 Probability Lecture 7: Random Variable and Expectation Pengyuan (Penelope) Wang June 2, 2011 Review Properties of Probability Conditional Probability The Law of Total Probability Bayes Formula

More information

CH 3 P1. Two fair dice are rolled. What is the conditional probability that at least one lands on 6 given that the dice land on different numbers?

CH 3 P1. Two fair dice are rolled. What is the conditional probability that at least one lands on 6 given that the dice land on different numbers? CH 3 P1. Two fair dice are rolled. What is the conditional probability that at least one lands on 6 given that the dice land on different numbers? P7. The king comes from a family of 2 children. what is

More information

Lecture 10. Variance and standard deviation

Lecture 10. Variance and standard deviation 18.440: Lecture 10 Variance and standard deviation Scott Sheffield MIT 1 Outline Defining variance Examples Properties Decomposition trick 2 Outline Defining variance Examples Properties Decomposition

More information

MATH1231 Algebra, 2017 Chapter 9: Probability and Statistics

MATH1231 Algebra, 2017 Chapter 9: Probability and Statistics MATH1231 Algebra, 2017 Chapter 9: Probability and Statistics A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra

More information

ELEG 3143 Probability & Stochastic Process Ch. 1 Probability

ELEG 3143 Probability & Stochastic Process Ch. 1 Probability Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 1 Probability Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Applications Elementary Set Theory Random

More information

Denker FALL Probability- Assignment 6

Denker FALL Probability- Assignment 6 Denker FALL 2010 418 Probability- Assignment 6 Due Date: Thursday, Oct. 7, 2010 Write the final answer to the problems on this assignment attach the worked out solutions! Problem 1: A box contains n +

More information

Discrete Probability

Discrete Probability MAT 258 Discrete Mathematics Discrete Probability Kenneth H. Rosen and Kamala Krithivasan Discrete Mathematics 7E Global Edition Chapter 7 Reproduced without explicit consent Fall 2016 Week 11 Probability

More information

Discrete random variables and probability distributions

Discrete random variables and probability distributions Discrete random variables and probability distributions random variable is a mapping from the sample space to real numbers. notation: X, Y, Z,... Example: Ask a student whether she/he works part time or

More information

CSC Discrete Math I, Spring Discrete Probability

CSC Discrete Math I, Spring Discrete Probability CSC 125 - Discrete Math I, Spring 2017 Discrete Probability Probability of an Event Pierre-Simon Laplace s classical theory of probability: Definition of terms: An experiment is a procedure that yields

More information

Exam III Review Math-132 (Sections 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 8.1, 8.2, 8.3)

Exam III Review Math-132 (Sections 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 8.1, 8.2, 8.3) 1 Exam III Review Math-132 (Sections 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 8.1, 8.2, 8.3) On this exam, questions may come from any of the following topic areas: - Union and intersection of sets - Complement of

More information

Probability deals with modeling of random phenomena (phenomena or experiments whose outcomes may vary)

Probability deals with modeling of random phenomena (phenomena or experiments whose outcomes may vary) Chapter 14 From Randomness to Probability How to measure a likelihood of an event? How likely is it to answer correctly one out of two true-false questions on a quiz? Is it more, less, or equally likely

More information

Probability 5-4 The Multiplication Rules and Conditional Probability

Probability 5-4 The Multiplication Rules and Conditional Probability Outline Lecture 8 5-1 Introduction 5-2 Sample Spaces and 5-3 The Addition Rules for 5-4 The Multiplication Rules and Conditional 5-11 Introduction 5-11 Introduction as a general concept can be defined

More information

Maths-III. Important Types in Maths III. Prepared By : Sameer V. shaikh { }

Maths-III. Important Types in Maths III. Prepared By : Sameer V. shaikh { } Mhs-III Important Types in Mhs III Prepared By : Sameer V. shaikh {Engr.sameer@gmail.com} {9765158158} MINIMUM Imp TYPES FOR MATHS III Types of Problems No Type of Problem Min/max marks Locion in Q.P

More information

CDA6530: Performance Models of Computers and Networks. Chapter 2: Review of Practical Random Variables

CDA6530: Performance Models of Computers and Networks. Chapter 2: Review of Practical Random Variables CDA6530: Performance Models of Computers and Networks Chapter 2: Review of Practical Random Variables Two Classes of R.V. Discrete R.V. Bernoulli Binomial Geometric Poisson Continuous R.V. Uniform Exponential,

More information

EXAM. Exam #1. Math 3342 Summer II, July 21, 2000 ANSWERS

EXAM. Exam #1. Math 3342 Summer II, July 21, 2000 ANSWERS EXAM Exam # Math 3342 Summer II, 2 July 2, 2 ANSWERS i pts. Problem. Consider the following data: 7, 8, 9, 2,, 7, 2, 3. Find the first quartile, the median, and the third quartile. Make a box and whisker

More information

(b). What is an expression for the exact value of P(X = 4)? 2. (a). Suppose that the moment generating function for X is M (t) = 2et +1 3

(b). What is an expression for the exact value of P(X = 4)? 2. (a). Suppose that the moment generating function for X is M (t) = 2et +1 3 Math 511 Exam #2 Show All Work 1. A package of 200 seeds contains 40 that are defective and will not grow (the rest are fine). Suppose that you choose a sample of 10 seeds from the box without replacement.

More information

Probability, For the Enthusiastic Beginner (Exercises, Version 1, September 2016) David Morin,

Probability, For the Enthusiastic Beginner (Exercises, Version 1, September 2016) David Morin, Chapter 8 Exercises Probability, For the Enthusiastic Beginner (Exercises, Version 1, September 2016) David Morin, morin@physics.harvard.edu 8.1 Chapter 1 Section 1.2: Permutations 1. Assigning seats *

More information

Introduction to Probability, Fall 2013

Introduction to Probability, Fall 2013 Introduction to Probability, Fall 2013 Math 30530 Section 01 Homework 4 Solutions 1. Chapter 2, Problem 1 2. Chapter 2, Problem 2 3. Chapter 2, Problem 3 4. Chapter 2, Problem 5 5. Chapter 2, Problem 6

More information

STAT 345 Spring 2018 Homework 4 - Discrete Probability Distributions

STAT 345 Spring 2018 Homework 4 - Discrete Probability Distributions STAT 345 Spring 2018 Homework 4 - Discrete Probability Distributions Name: Please adhere to the homework rules as given in the Syllabus. 1. Coin Flipping. Timothy and Jimothy are playing a betting game.

More information

Lecture 10: Bayes' Theorem, Expected Value and Variance Lecturer: Lale Özkahya

Lecture 10: Bayes' Theorem, Expected Value and Variance Lecturer: Lale Özkahya BBM 205 Discrete Mathematics Hacettepe University http://web.cs.hacettepe.edu.tr/ bbm205 Lecture 10: Bayes' Theorem, Expected Value and Variance Lecturer: Lale Özkahya Resources: Kenneth Rosen, Discrete

More information

Probability Theory and Random Variables

Probability Theory and Random Variables Probability Theory and Random Variables One of the most noticeable aspects of many computer science related phenomena is the lack of certainty. When a job is submitted to a batch oriented computer system,

More information

STAT 516 Midterm Exam 2 Friday, March 7, 2008

STAT 516 Midterm Exam 2 Friday, March 7, 2008 STAT 516 Midterm Exam 2 Friday, March 7, 2008 Name Purdue student ID (10 digits) 1. The testing booklet contains 8 questions. 2. Permitted Texas Instruments calculators: BA-35 BA II Plus BA II Plus Professional

More information

STAT 418: Probability and Stochastic Processes

STAT 418: Probability and Stochastic Processes STAT 418: Probability and Stochastic Processes Spring 2016; Homework Assignments Latest updated on April 29, 2016 HW1 (Due on Jan. 21) Chapter 1 Problems 1, 8, 9, 10, 11, 18, 19, 26, 28, 30 Theoretical

More information

Unit II. Page 1 of 12

Unit II. Page 1 of 12 Unit II (1) Basic Terminology: (i) Exhaustive Events: A set of events is said to be exhaustive, if it includes all the possible events. For example, in tossing a coin there are two exhaustive cases either

More information

Random Variables Example:

Random Variables Example: Random Variables Example: We roll a fair die 6 times. Suppose we are interested in the number of 5 s in the 6 rolls. Let X = number of 5 s. Then X could be 0, 1, 2, 3, 4, 5, 6. X = 0 corresponds to the

More information

Ch 2: Probability. Contents. Fall 2017 UAkron Dept. of Stats [3470 : 451/551] Theoretical Statistics I

Ch 2: Probability. Contents. Fall 2017 UAkron Dept. of Stats [3470 : 451/551] Theoretical Statistics I Fall 2017 UAkron Dept. of Stats [3470 : 451/551] Theoretical Statistics I Ch 2: Probability Contents 1 Preliminaries 3 1.1 Interpretation of Probability (2.2)......................................................

More information

II. The Binomial Distribution

II. The Binomial Distribution 88 CHAPTER 4 PROBABILITY DISTRIBUTIONS 進佳數學團隊 Dr. Herbert Lam 林康榮博士 HKDSE Mathematics M1 II. The Binomial Distribution 1. Bernoulli distribution A Bernoulli eperiment results in any one of two possible

More information

Mutually Exclusive Events

Mutually Exclusive Events 172 CHAPTER 3 PROBABILITY TOPICS c. QS, 7D, 6D, KS Mutually Exclusive Events A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes

More information

IEOR 3106: Introduction to Operations Research: Stochastic Models. Professor Whitt. SOLUTIONS to Homework Assignment 1

IEOR 3106: Introduction to Operations Research: Stochastic Models. Professor Whitt. SOLUTIONS to Homework Assignment 1 IEOR 3106: Introduction to Operations Research: Stochastic Models Professor Whitt SOLUTIONS to Homework Assignment 1 Probability Review: Read Chapters 1 and 2 in the textbook, Introduction to Probability

More information

Stochastic Models of Manufacturing Systems

Stochastic Models of Manufacturing Systems Stochastic Models of Manufacturing Systems Ivo Adan Organization 2/47 7 lectures (lecture of May 12 is canceled) Studyguide available (with notes, slides, assignments, references), see http://www.win.tue.nl/

More information

Name: Firas Rassoul-Agha

Name: Firas Rassoul-Agha Midterm 1 - Math 5010 - Spring 016 Name: Firas Rassoul-Agha Solve the following 4 problems. You have to clearly explain your solution. The answer carries no points. Only the work does. CALCULATORS ARE

More information