Probabilistic System Analysis: Negative Binomial and Geometric Distributions

Probabilistic System Analysis Lecture 7 Dr.-Ing. Erwin Sitompul President University http://zitompul.wordpress.com 2 0 2 1 President University Erwin Sitompul PSA 7/1

Chapter 5.5 Negative Binomial and Geometric Distributions Negative Binomial Distribution Consider an experiment where the properties are the same as those listed for a binomial experiment, with the exception that the trials will be repeated until a fixed number of successes occur. We are interested in the probability that the kth success occurs on the xth trial. This kind of experiment is called negative binomial experiment. The number X of trials to produce k successes in a negative binomial experiment is called a negative binomial random variable, and its probability distribution is called the negative binomial distribution. |Negative Binomial Distribution| If repeated independent trials can result in a success with probability p and a failure with probability q=1 p, then the probability distribution of the random variable X, the number of the trial on which the kth success occurs, is x k = = + + * k ( ; , ) b x k p , , 1, 2,... C p q x k k k 1 1 x k President University Erwin Sitompul PSA 7/2

Chapter 5.5 Negative Binomial and Geometric Distributions Negative Binomial Distribution In an NBA (National Basketball Association) championship series, the team which wins four games out of seven will be the winner. Suppose that team A has probability 0.55 of winning over the team B and both teams A and B face each other in the championship games. (a) What is the probability that team A will win the series in six games? (b) What is the probability that team A will win the series? 6 4 = = * 4 (6;4,0.55) (0.55) (0.45) b C 0.1853 (a) 6 1 4 1 (b) ( ) P team A wins the championship series b = + = + + = + + * * * * (4;4,0.55) 0.0915 0.1647 0.1853 0.1668 0.6083 (5;4,0.55) + (6;4,0.55) (7;4,0.55) Team A wins b b b President University Erwin Sitompul PSA 7/3

Chapter 5.5 Negative Binomial and Geometric Distributions Negative Binomial Distribution In an NBA (National Basketball Association) championship series, the team which wins four games out of seven will be the winner. Suppose that team A has probability 0.55 of winning over the team B and both teams A and B face each other in the championship games. (c) If both teams face each other in a regional playoff series and the winner is decided by winning three out of five games, what is the probability that team A will win a playoff? Team A wins (c) ( = ) P team A wins the regional series + + * * (3;3,0.55) (5;3,0.55) 0.1664 0.2246 0.2021 + (4;3,0.55) b b b * = + = 0.5931 Team B wins President University Erwin Sitompul PSA 7/4

Chapter 5.5 Negative Binomial and Geometric Distributions Geometric Distribution If we consider the special case of the negative binomial distribution where k=1, we have a probability distribution for the number of trials required for a single success. If repeated independent trials can result in a success with probability p and a failure with probability q=1 p, then the probability distribution of the random variable X, the number of the trial on which the first success occurs, is 1 ( ; ) , 1,2,3,... g x p pq x = = x The mean and variance of a random variable following the geometric distribution are 1 1 p p p = = 2 and 2 President University Erwin Sitompul PSA 7/5

Chapter 5.5 Negative Binomial and Geometric Distributions Geometric Distribution In a certain manufacturing process it is known that, on the average, 1 in every 100 items is defective. What is the probability that the fifth item inspected is the first defective item found? = = 4 (5;0.01) (0.01)(0.99) 0.0096 g At busy time a telephone exchange is very near capacity, so callers have difficulty placing their calls. It may be of interest to know the number of attempts necessary in order to gain a connection. Suppose that we let p = 0.05 be the probability of a connection during busy time. We are interested in knowing the probability that 5 attempts are necessary for a successful call. = = = = ( 5) (5;0.05) (0.05)(0.95) P X g 4 0.041 President University Erwin Sitompul PSA 7/6

Chapter 5.6 Poisson Distribution and Poisson Process Poisson Distribution and Poisson Process Experiments yielding numerical values of a random variable X, the number of outcomes occurring during a given time interval or in a specified space, are called Poisson experiments. The time interval may be given in any length, such as minute, day, week, and month. The specified space may be a line segment, an area, a volume, or a piece of material. President University Erwin Sitompul PSA 7/7

Chapter 5.6 Poisson Distribution and Poisson Process Properties of Poisson Process A Poisson experiment is derived from the Poisson process and possesses the following properties: 1. The number of outcomes occurring in one time period or one space is independent of the number that occurs in any other disjoint time period or space. 2. The average probability that a single outcome will occur during a certain time period or in a certain space is constant. 3. The probability that more than one outcome will occur at the same time period or the same space is negligible. Poisson process: average in time or average in space between events is known. But, they are randomly spaced. For example: The average that a driver will be fined because he drives through red traffic light is 0.1/month. He can get caught by the police because he drives through red traffic light for 2 months in a row. But he also could do the same violation without any warning for 4 years. President University Erwin Sitompul PSA 7/8

Chapter 5.6 Poisson Distribution and Poisson Process Poisson Distribution and Poisson Process |Poisson Distribution| The probability distribution of the Poisson random variable X, representing the number of outcomes occurring in a given time interval or specified space denoted by t, is ( ) ( ; ) , 0,1,2,... ! x where is the average number of outcomes per unit time or space, and e = 2.71828.... (natural number). t x e t = = p x t x The mean and variance of the Poisson distribution p(x; t) both have the value t. President University Erwin Sitompul PSA 7/9

Chapter 5.6 Poisson Distribution and Poisson Process Poisson Distribution and Poisson Process During a laboratory experiment the average number of radioactive particles passing through a counter in 1 millisecond is 4. What is the probability that 6 particles enter the counter in a given millisecond? t = = = 6, 4 1 4 x Table A.2 gives help 4 6 (4) 6! e = = (6;4) p 0.1042 Ten is the average number of oil tankers arriving each day at a certain port city. The facilities at the port can handle at most 15 tankers per day. What is the probability that on a given day tankers have to be turned away? Table A.2 gives help 15 = = ( 15) 1 ( 15) 1 ( ;10) p x P X P X = 0 e x 10 1 10 2 10 15 (10) 1! (10) 2! (10) 15! e e = + + + 1 1 0.9513 = = 0.0487 President University Erwin Sitompul PSA 7/10

Chapter 5.6 Poisson Distribution and Poisson Process Table A.2 Poisson Probability Sums President University Erwin Sitompul PSA 7/11

Chapter 5.6 Poisson Distribution and Poisson Process Poisson Distribution As a Limit of Binomial It should be clear from the three properties of the Poisson process that the Poisson distribution relates to the binomial distribution. (Success or Failure at a certain time or space). In the case of the binomial, if n is quite large and p is small, the conditions begin to simulate the continuous time or space implications of the Poisson process. Poisson distribution can be taken as a limiting form of the binomial distribution when n and p 0, and np remains constant. If the conditions are fulfilled, the Poisson distribution can be used with = np, to approximate binomial distribution. Let X be a binomial random variable with probability distribution b(x;n,p). When n and p 0, and = np remains constant, ( ; , ) ( , ) b x n p p x President University Erwin Sitompul PSA 7/12

Chapter 5.6 Poisson Distribution and Poisson Process Poisson Distribution and Poisson Process In a certain industrial facility accidents occur infrequently. It is known that the probability of an accident on any given day is 0.005 and accidents are independent of each other. (a) What is the probability that in any given period of 400 days there will be an accident on one day? (b) What is the probability that there are at most three days with an accident? = t = = 1, (0.005)(400) 2 x 2 1 (2) 1! e Considered as Poisson process = = (a) (1;2) p 0.2707 = 1 399 = (1;400,0.005) (0.005) (0.995) b C 0.2707 400 1 Considered as Bernoulli process 2 x 3 3 (2) ! x e (b) = = = ( 3) ( ;2) p x P X 0.8571 = = 0 0 x 3 x = = ( 3) ( ;400,0.005) b x P X 0.8571 = 0 x President University Erwin Sitompul PSA 7/13

Chapter 5.6 Poisson Distribution and Poisson Process Poisson Distribution and Poisson Process In a manufacturing process where glass products are produced, defects or bubbles occur, occasionally rendering the piece undesirable for marketing. It is known that, on average, 1 in every 1000 of these items produced has one or more bubbles. What is the probability that a random sample of 8000 will yield fewer than 7 items possessing bubbles? = = (8000)(0.001) 8 6 6 = = ( 7) ( ;8000,0.001) b x ( ;8) p x P X 0.3134 = = 0 0 x x Actually a problem for Binomial Distribution Solved by approximation using Poisson Distribution President University Erwin Sitompul PSA 7/14

Probabilistic System Analysis Homework 7A 1. A communications system consists of n components, each of which will, independently, function with probability p. The total system will be able to operate effectively if at least one-half of its components function. For what values of p is a 5-component system more likely to operate effectively than a 3-component system? (Ro.E5.1c s144) 2. It has been established that the number of defective stereos produced daily at a certain plant is Poisson distributed with mean 4. Over a 2-day span, what is the probability that the number of defective stereos does not exceed 3? (Ro.E5.2f s+10) 3. A mining company conducts a series of geological survey to find suitable mining sites. The probability that a survey indicates a location is suitable is 30%. (a) Determine the probability that the first suitable mining site will be found after 7 surveys. (b) What is the probability that from the first 5 surveys, 3 suitable mining site will be found? (c) What is the probability that the 3rd suitable mining site will be found after conducting 5 surveys? President University Erwin Sitompul PSA 7/15