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EVERYDAY ENGINEERING EXAMPLES FOR SIMPLE CONCEPTS ENGR 3200 – Probability and Statistics Dr. Heriberto Barriera Copyright © 2015 Discrete Probability Distribution – Negative Binomial MSEIP – Engineering Everyday Engineering Examples An Easy Way to Demonstrate Negative Binomial Distribution Engage: In a small group, roll a die until a “1” is observed. Repeat this process a total of 30 times and record the number of rolls required to get a “1” in each of the 30 trials of the experiment. Trial # 1 2 3 4 5 6 7 8 9 10 Number of rolls to get a “1” Trial # Number of rolls to get a “1” 11 12 13 14 15 16 17 18 19 20 Trial # Number of rolls to get a “1” 21 22 23 24 25 26 27 28 29 30 Use your results to obtain a point estimate of the mean and compare the results with the expected value of the number of rolls of a die required to obtain a “1” using the formula for the negative binomial distribution. Explore: The sample space of an experiment, denoted S, is the set of all possible outcomes Sample Space: S ={1, 2, 3, 4, 5, 6} Page Outcomes: landing with a 1, 2, 3, 4, 5, or 6 face up. 1 of that experiment. The number of possible outcomes when rolling a die are: You can begin rolling a die until a “1” is observed. As you can see, the number of times necessary until a “1” is observed, is different. The probability to get a “1” in a fair die is 1/6. As the number of time rolling a die is increased, the average of times to get a ”1” is approximated. Explain: A negative binomial experiment is a statistical experiment that has the following properties: The experiment consists of x repeated trials. Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure. The probability of success, denoted by P, is the same on every trial. The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials. The experiment continues until r successes are observed, where r is specified in advance. Elaborate: This experiment is a negative binomial experiment because: The experiment consists of repeated trials. We roll a die repeatedly until it a “1” is observed. Each trial can result in just two possible outcomes – success “1” or failure “2,3,4,5,6”. The probability of success is constant – 1/6 on every trial. The trials are independent; that is, getting a “1”on one trial does not affect The experiment continues until a fixed number of successes have occurred; in this case, a “1”. Page 2 whether we get a “1” on other trials. The pmf of the negative binomial rv X with parameters r = number of S’s and p = P(S) is x r 1 r p 1 p x nb( x; rp) r 1 Where x = 0, 1, 2, … and represent the number of failures necessary to get the r success The expected value and the variance for the negative binomial distribution is E( X ) r (1 p) p V (X ) r (1 p) p2 Evaluate: Invite students to attempt the following problem: Example: A basketball player made 45% of his shots from the three point line. Find the probability that this player will get his 5 made on the 9 attempt. Using the negative binomial distribution: x = 4, that represents the number of failures p = 0.45, that represents the probability of success Page x r 1 r p 1 p x nb( x; rp) r 1 3 r = 5, that represents the number of success 4 5 1 8 (.45)5 (1 .45) 4 (.45)5 (1 .45) 4 0.1182 nb(4;5,.45) 5 1 4 What is the expected number of attempts to get the 5 made shots? E ( x) r (1 p) p E ( x) 5(1 .45) 6 .45 Therefore the number of attempts necessary to get 5 shots made are 11 (5 (the number Page 4 of success) + 6 (the number of failures)).