Unformatted text preview: Ch. 5: Discrete Random Variables and Their Distribu9ons Random Variable: Example We are playing a very simplified version of blackjack in which each person is only dealt 2 cards. We are interested in the sum of the cards a) is the sum of the cards a quan9ta9ve or qualita9ve variable? b) Is this a random variable? c) What is the sample space for this random variable? d) what are the possible values for the random variable? Random Variable Discrete: Example Supposed that you draw 3 cards from a deck of cards and record whether the suit is black or red. Let X be the total number of red cards. a) construct a table that shows the values of X b) Explain why X is a discrete random variable Iden9fy the following events in words and as a subset of the sample space. c) {X = 2} d) {X 2} e) {X 1} PMF: Example Supposed that you draw 3 cards from a deck of cards and record whether the suit is black or red. Let X be the total number of red cards. a) What is pX(2)? b) Determine the PMF of X. c) Construct a probability histogram for X. Histogram for PMF Example px(x) 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0 1 2 3 x PMF: Example (cont) Supposed that you draw 3 cards from a deck of cards (with replacement) and record whether the suit is black or red. Let X be the total number of red cards. Determine the PMF when a) there are equal numbers of red and black cards. b) if out of 100 cards, 30 are red and 70 are black. c) for p red cards out of 100 cards. Outcome Probability p = 0.5 p = 0.3 General p RRR RRB RBR RBB BRR BRB BBR BBB PMF Example (cont) Outcome Probability p = 0.5 p = 0.3 General p RRR 1/8 = 0.125 RRB RBR RBB BRR BRB BBR BBB PMF Example (cont) Outcome Probability p = 0.5 p = 0.3 General p RRR 1/8 = 0.125 0.027 RRB RBR RBB BRR BRB BBR BBB PMF Example (cont) Outcome Probability p = 0.5 p = 0.3 General p RRR 1/8 = 0.125 0.027 p3(1 p)0 RRB RBR RBB BRR BRB BBR BBB PMF Example (cont) Outcome Probability p = 0.5 RRR 1/8 = 0.125 RRB 1/8 = 0.125 RBR 1/8 = 0.125 RBB 1/8 = 0.125 BRR 1/8 = 0.125 BRB 1/8 = 0.125 BBR 1/8 = 0.125 BBB 1/8 = 0.125 PMF Example (cont) p = 0.3 0.027 0.063 0.063 0.147 0.063 0.147 0.147 0.343 General p p3(1 p)0 p2(1 p)1 p2(1 p)1 p1(1 p)2 p2(1 p)1 p1(1 p)2 p1(1 p)2 p0(1 p)3 Histogram: interpreta9on of PMF px(x) 0.40 0.30 0.20 0.10 0.00 0 1 2 3 Theore9cal 0.40 0.30 0.20 0.10 0.00 x 0 1 2 3 Simulated x 1000 9mes 0.40 0.30 0.20 0.10 0.00 0 1 2 3 Simulated 10,000 9mes x Bernoulli Trials In each case, iden9fy the random experiment, the trials, a success, a failure, the success probability, and the failure probability. Explain why the trials can be considered Bernoulli trials or not. 1. Rolling a fair 4sided die and observing whether the number showing is a 1 or not. 2. The number of births of girls in a county hospital on any specific day. 3. If different pa9ents with the same condi9on take the same drug to see if it is effec9ve or not. 4. In a drug trial, some pa9ents with the same condi9on are given a drug and some are given a placebo to see if the drug is effec9ve or not. Bernoulli Trials (cont) In each case, iden9fy the random experiment, the trials, a success, a failure, the success probability, and the failure probability. Explain why the trials can be considered Bernoulli trials or not. 5. In quality control we want to see if a par9cular product is `bad'. We take random samples from an assembly line that uses different machines to product the product. 6. We look at the percentage that a basketball player makes her shots. Example 5.11: Bernoulli Trials Assume that the popula9on consists of N objects where each member can be considered having or not having an airibute. a) Random sampling with replacement: Suppose n members are randomly selected one at a 9me, with the property noted and then returned to popula9on for reselec9on. b) Random sampling without replacement. Suppose n members are randomly selected one at a 9me, with the property noted and then NOT returned to popula9on for reselec9on. Binomial r.v. A restaurant serves eight entres of fish, 12 of beef, and 10 of poultry. If customers select from these entres randomly, what is the probability that a) two of the next four customers order fish entres? b) at most one of the next four customers orders fish? c) at least one of the next four customers orders fish? Histogram Binomial Thm. X 0 1 2 3 4 else pX(x) 0.289 0.421 0.229 0.056 0.005 0 px(x) 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0 1 2 3 4 x Probability histograms for binomial distribu9ons with different p's px(x) 0.2 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0 1 2 3 4 5 6 7 8 x px(x) 0.5 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0 1 2 3 4 5 6 7 8 x px(x) 0.8 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0 1 2 3 4 5 6 7 8 x Probability histograms for binomial distribu9ons with different p's (cont) px(x) 0.5 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0 1 2 3 4 5 6 7 8 x px(x) 0.5 0.30 0.25 0.20 0.15 0.10 0.05 0.00 x 0 1 2 3 4 5 6 7 8 9 Hypergeometric r.v. A textbook author is preparing an answer key for the answers in a book. In 500 problems, the author has made 25 errors. If a second person checks seven of these calcula9ons randomly, what is the probability that he will detect two errors? Assume that the second person will definitely find the error in an incorrect answer. Hypergeometric r.v. Example 5.15* The Loio. In the Hoosier loio, a player specifies six numbers of her choice from the numbers 1 48. In the loiery drawing, six winning numbers are chosen at random without replacement from the numbers 1 48. To win a prize, a loio 9cket must contain three or more of the winning numbers. a) Determine the PMF of the r.v. X, the number of winning numbers on the player's 9cket. b) If the player buys one Loio 9cket, determine the probability that she wins a prize (at least 3 numbers correct). c) If the player buys one Loio 9cket per week for a year, determine the probability that she wins a prize at least once in the 52 tries. x Hypergeometric r.v. Example 5.15* The Loio. In the Hoosier loio, a player specifies six numbers of her choice from the numbers 1 48. In the loiery drawing, six winning numbers are chosen at random without replacement from the numbers 1 48. To win a prize, a loio 9cket must contain three or more of the winning numbers. These are the odds from the Hoosier loiery: 6 OF 6 1:12,271,512 5 OF 6 1:48,696 4 OF 6 1:950.18 3 OF 6 1:53.45 x Hypergeometric r.v. Example 5.16 Es9ma9ng the Size of a Popula9on. Suppose that an unknown number, N, of animals inhabit a region and that we want to es9mate that number. One procedure for doing so, onen referred to as the capturerecapture method, is to proceed as follows: 1. Capture M of the animals, mark them in some way, and then release the animals back into the region and give them 9me to disperse. 2. Capture n of the animals and note the number X that are marked that is, the number that are recaptures. 3. Use the data to es9mate N. What is N when M = 50, n = 100, X = 3? x Examples of Poisson approxima9on to the Binomial Distribu9on 1. The number of misprints on a page of a book. 2. The number of people in a community living to 100 years of age. 3. The number of wrong telephone numbers that are dialed in a day. 4. The number of packages of cat treats sold in a par9cular store each day. 5. The number of vacancies occurring during a year in the Supreme Court. x Example: Poisson Approxima9on to a Binomial On my page of notes, I have 2150 characters. Say that the chance of a typo (aner I proof it) is 0.001. a) What is the probability of exactly 1 typo on this page? b) What is the probability of at most 3 typos? x Poisson Approxima9on to a Binomial x More Examples of Poisson R.V.'s 6. The number of pa9ents that arrive in an emergency room (or any other loca9on) between 6:00 pm and 7:00 pm (or any other period of 9me). 7. The number of alpha par9cles emiied per minute by a radioac9ve substance. 8. The number of cars that are located on a par9cular sec9on of highway at a given 9me. x Example: Poisson r.v. Between the hours of 2 and 4 pm, If the number of phone calls per minute coming into the switchboard of a company P(2.5). Find the probability that the number of calls during one par9cular minute will be a) 0 b) 2 or fewer x Shape of Poisson PMF 0.30 = 2.5 px(x) 0.25 0.20 0.15 0.10 0.05 0.00 0 1 2 3 4 5 6 7 8 9 10 11 12 13 x x Shape of Example 5.19 = 6.9 x Examples of a Poisson Process 1. The number of pa9ents that arrive in an emergency room (or any other loca9on) between 6:00 pm and 7:00 pm (or any other period of 9me). 2. The number of alpha par9cles emiied per minute by a radioac9ve substance. 3. The number of cars that are located on a par9cular sec9on of highway at a given 9me. x Poisson Process: m x Every second, 1.8 cosmic rays hit a specific spot on earth. Assume that we start coun9ng at t = 0 seconds. a) What is the probability that there are exactly 12 cosmic rays hiqng the spot between 10 seconds and 15 seconds? b) Given that exactly 3 cosmic rays hit the spot between 4 seconds and 5 seconds, what is the probability that 12 cosmic rays hit the spot between 10 seconds 15 seconds? c) What is the probability that at least one cosmic ray hits the spot between 4 seconds and 5 seconds and between 10 seconds and 15 seconds? x Poisson Process: Example Example: Geometric r.v. Suppose that we roll an nsided die un9l a '1' is rolled. Let X be the number of 9mes it takes to roll the '1'. What is the PMF of X? x Shape of Geometric PMF px(x) 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0 1 2 3 4 5 6 7 8 9 10 0.4 x x Useful Geometric Formulas 1r r = 1  r r 1 j = 0
j n n+1 1 r = 1  r r< 1 j = 0 j r r j ar = 1  r r 1, a b j = b a b +1 ra j ar = 1  r r< 1 j = x Suppose that we roll an nsided die un9l a '1' is rolled. Let X be the number of 9mes it takes to roll the '1'. What is the PMF of X? Assume that n = 20. a) What is the probability that it will take exactly 10 rolls? b) What is the probability that it will take no more than 10 rolls? c) What is the probability that it will take between 10 and 20 rolls (exclusive)? d) Determine the number of rolls so that the person has a 90% or greater chance of rolling a `1'? x Example: Geometric r.v. (cont) Example: Geometric r.v. (cont) Suppose that we roll an nsided die un9l a '1' is rolled. Let X be the number of 9mes it takes to roll the '1'. What is the PMF of X? Assume that n = 20. a) What is the probability that it will take exactly 10 rolls? b) Assuming that it takes more than 20 rolls to roll the `1'. Find the probability that it will take 10 more rolls to roll the `1'? x Example: Discrete Uniform (Example 5.24) A charitable organiza9on is conduc9ng a raffle in which the grand prize is a new car. Five thousand 9ckets, numbered 0001, 0002, ..., 5000 are sold at $10 each. At the grandprize drawing, one 9cket stub will be selected at random from the 5000 9cket stubs. Let X denote the number on the 9cket stub obtained. a) Find the PMF of the r.v. X. b) Suppose that you hold 9ckets numbered 1003 1025. Express the event that you win the grand prize in terms of the random variable X, and then compute the probability of the event. x Example: Nega9ve Binomial r.v. Suppose that we roll an nsided die un9l a '1' is rolled. Let X be the number of 9mes it takes to roll the ninth '1'. What is the PMF of X? x Every second, 1.8 cosmic rays hit a specific spot on earth. Assume that we start coun9ng at t = 0 seconds. a) What is the probability that there are exactly 12 cosmic rays hiqng the spot between 10 seconds and 15 seconds? b) Given that exactly 3 cosmic rays hit the spot between 4 seconds and 5 seconds, what is the probability that 12 cosmic rays hit the spot between 10 seconds 15 seconds? c) What is the probability that at least one cosmic ray hits the spot between 4 seconds and 5 seconds and between 10 seconds and 15 seconds? x Example 1 The Loio. In the Hoosier loio, a player specifies six numbers of her choice from the numbers 1 48. In the loiery drawing, six winning numbers are chosen at random without replacement from the numbers 1 48. To win a prize, a loio 9cket must contain three or more of the winning numbers. a) Determine the PMF of the r.v. X, the number of winning numbers on the player's 9cket. b) If the player buys one Loio 9cket, determine the probability that she wins a prize (at least 3 numbers correct). c) If the player buys one Loio 9cket per week for a year, determine the probability that she wins a prize at least once in the 52 tries. x Example 2 Example 3 A charitable organiza9on is conduc9ng a raffle in which the grand prize is a new car. Five thousand 9ckets, numbered 0001, 0002, ..., 5000 are sold at $10 each. At the grandprize drawing, one 9cket stub will be selected at random from the 5000 9cket stubs. Let X denote the number on the 9cket stub obtained. a) Find the PMF of the r.v. X. b) Suppose that you hold 9ckets numbered 1003 1025. Express the event that you win the grand prize in terms of the random variable X, and then compute the probability of the event. x Example 4 Between the hours of 2 and 4 pm, If the number of phone calls per minute coming into the switchboard of a company with parameter 2.5. Find the probability that the number of calls during one par9cular minute will be a) 0 b) 2 or fewer x Example 5 On my page of notes, I have 2150 characters. Say that the chance of a typo (aner I proof it) is 0.001. a) What is the probability of exactly 1 typo on this page? b) What is the probability of at most 3 typos? x Example 6 Suppose that we roll an nsided die un9l a '1' is rolled. Let X be the number of 9mes it takes to roll the ninth '1'. What is the PMF of X? x Example 7 A textbook author is preparing an answer key for the answers in a book. In 500 problems, the author has made 25 errors. If a second person checks seven of these calcula9ons randomly, what is the probability that he will detect two errors? Assume that the second person will definitely find the error in an incorrect answer. x Example 8 Suppose that we
roll an nsided die un9l a '1' is rolled. Let X be the number of 9mes it takes to roll the '1'. What is the PMF of X? x Example 9 A restaurant serves eight entres of fish, 12 of beef, and 10 of poultry. If customers select from these entres randomly, what is the probability that a) two of the next four customers order fish entres? b) at most one of the next four customers orders fish? c) at least one of the next four customers orders fish? x ...
View
Full Document
 Spring '08
 Staff
 Probability, Probability theory, Grammatical number, Roll, loio

Click to edit the document details