Lesson 4 Discrete Probability Distributions Kafu Wong 2004

Outline Random variables and probability distribution Features of univariate probability distribution Features of bivariate probability distribution Marginal distribution and Conditional distribution Expectation and conditional expectation Variance, Covariance and Correlation Coefficient Binomial Probability Distribution Hypergeometric Probability Distribution Poisson Probability Distribution Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 2

Random Variables and probability distribution n A random variable is a numerical value determined by the outcome of an experiment. A random variable is often denoted by a capital letter, e. g. , X or Y. n A probability distribution is the listing of all possible outcomes of an experiment and the corresponding probability. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 3

Types of Probability Distributions n A discrete probability distribution can assume only certain outcomes (need not be finite) – for random variables that take discrete values. n The number of students in a class. n The number of children in a family. n A continuous probability distribution can assume an infinite number of values within a given range – for random variables that take continuous values. n The distance students travel to class. n The time it takes an executive to drive to work. n The amount of money spent on your last haircut. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 4

Types of Probability Distributions Probability distribution may be classified according to the number of random variables it describes. Number of random variables Ka-fu Wong © 2004 Joint distribution 1 Univariate probability distribution 2 Bivariate probability distribution 3 Trivariate probability distribution … … n Multivariate probability distribution ECON 1003: Analysis of Economic Data 5

Features of a Univariate Discrete Distribution n Let x 1, …, x. N be the list of all possible outcomes (N of them). n The main features of a discrete probability distribution are: n The probability of a particular outcome, P(xi), is between 0 and 1. 00. n The sum of the probabilities of the various outcomes is 1. 00. That is, P(x 1) + … + P(x. N) = 1 n The outcomes are mutually exclusive. That is, P(x 1 and x 2) = 0 and Outcome Prob. P(x 1 or x 2) = P(x 1)+ P(x 2) x 1 P(x 1) n Generally, for all i not equal to k. x 2 P(x 2) P(xi and xk) = 0. … … P(xi or xk) = P(xi)+ P(xk) x. N Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data P(x. N) 6

Features of a Univariate Discrete Distribution Can the following be a probability distribution of a random variable? x Prob. 1 0. 2 2 0. 3 1 0. 6 3 0. 1 2 0. 3 1 0. 4 3 0. 1 Ka-fu Wong © 2004 event Prob. 1 or 2 0. 6 2 or 3 0. 3 3 or 1 0. 1 ECON 1003: Analysis of Economic Data 7

EXAMPLE: Univariate probability distribution n Consider a random experiment in which a coin is tossed three times. Let x be the number of heads. Let H represent the outcome of a head and T the outcome of a tail. n The possible outcomes for such an experiment will be: TTT, TTH, THT, THH, HTT, HTH, HHT, HHH. n Thus the possible values of x (number of heads) are x=0: x=1: x=2: x=3: TTT TTH, THT, HTT THH, HTH, HHT HHH P(x=0) =1/8 P(x=1) =3/8 P(x=2) =3/8 P(x=3) =1/8 If the coin is fair n From the definition of a random variable, x as defined in this experiment, is a random variable. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 8

Features of a Bivariate Discrete Distribution n If X and Y are discrete random variables, we may define their joint probability function as PXY(xi, yi) n Let (x 1, …, x. R) and (y 1, …, y. S) be the list of all possible outcomes for X and Y respectively. n The main features of a bivariate discrete probability distribution are: n The probability of a particular outcome, PXY(xi, yi) is between 0 and 1. n The sum of the probabilities of the various outcomes is 1. 00. That is, PXY(x 1, y 1) + PXY(x 2, y 1) +…+ PXY(x. R, y 1) + + … + PXY(x. R, y. S) = 1 n The outcomes are mutually exclusive. That is, if xi not equal to xk, or yi not equal to yk PXY((xi, yi) and (xk, yk)) = 0 and PXY((xi, yi) or (xk, yk)) = PXY(xi, yi) + PXY(xk, yk) Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 9

Example: Bivariate Discrete Distribution X takes 3 possible values and Y takes 4 possible values. Ka-fu Wong © 2004 y 1 y 2 y 3 y 4 x 1 P(x 1, y 1) P(x 1, y 2) P(x 1, y 3) P(x 1, y 4) x 2 P(x 2, y 1) P(x 2, y 2) P(x 2, y 3) P(x 2, y 4) x 3 P(x 3, y 1) P(x 3, y 2) P(x 3, y 3) P(x 3, y 4) ECON 1003: Analysis of Economic Data 10

EXAMPLE: Bivariate distribution The joint distribution of the movement of Hang Seng Index (HSI) and weather is shown in the following table. Rainy Not Rainy Totals HSI falls 0. 15 0. 4 0. 55 HSI rises 0. 25 0. 45 0. 35 0. 65 1. 0 Totals Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 11

EXAMPLE: Bivariate distribution The joint distribution of the movement of Hang Seng Index (HSI) and weather is shown in the following table. Rainy Not Rainy HSI falls 0 a HSI rises 0 b Totals 0 Totals Suppose …. . PX|Y(x | y) = P(X = x | Y = y)=P(x, y)/P(y) if P(Y = y) > 0 PX|Y(x | y) =0 if P(Y = y) = 0 P(HSI falls|Rainy) = P(HSI falls, Rainy) / P(Rainy)= 0/0 Forcing P(HSI falls|Rainy) in the definition eliminates the difficulty in interpreting 0/0. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 12

Marginal Distributions Y X Rainy Not Rainy Totals HSI falls 0. 15 0. 4 0. 55 HSI rises 0. 25 0. 45 0. 35 0. 65 1. 0 Totals n The marginal probability function of X. PX(x) = y. PXY(x, y) = PXY(x, y 1) +PXY(x, y 2) +…+ PXY(x, yn) P(HSI falls)= P(HSI falls and rainy) + P(HSI falls and not rainy) P(HSI rises)= P(HSI rises and rainy) + P(HSI rises and not rainy) n The double sum x y. PXY(x, y) = P(HSI falls and rainy) + P(HSI falls and not rainy)+ P(HSI rises and rainy) + P(HSI rises and not rainy) = P(HSI falls)+P(HSI rises)= 1 Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 13

Marginal Distributions Y X Rainy Not Rainy Totals HSI falls 0. 15 0. 4 0. 55 HSI rises 0. 25 0. 45 0. 35 0. 65 1. 0 Totals n The marginal probability function of X. y. PXY(x, y) = PX(x). n The marginal probability function of Y. x. PXY(x, y) = PY(y). n The double sum y x. PXY(x, y) = 1 Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 14

Conditional Distributions Y X Rainy Not Rainy Totals HSI falls 0. 15 0. 4 0. 55 HSI rises 0. 25 0. 45 0. 35 0. 65 1. 0 Totals n The conditional probability function of X given Y: PX|Y(x | y) = P(X = x | Y = y) = PXY(x, y)/PY(y) if P(Y = y) > 0 PX|Y(x | y) =0 if P( Y = y) = 0 Note that PX|Y(x | y) when P(Y = y) = 0 is undefined using the top formula. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 15

Conditional Distributions Y X Rainy Not Rainy Totals HSI falls 0. 15 0. 4 0. 55 HSI rises 0. 25 0. 45 0. 35 0. 65 1. 0 Totals n For each fixed y this is a probability function for X, i. e. the conditional probability function is non-negative and XPX|Y(x | y) = PX|Y(x 1 | y)+ PX|Y(x 2 | y) = PX, Y(x 1, y)/ PY(y) + PX, Y(x 2, y)/ PY(y) =[PX, Y(x 1, y) + PX, Y(x 2, y)]/ PY(y) =1. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 16

Conditional Distributions Y X Rainy Not Rainy Totals HSI falls 0. 15 0. 4 0. 55 HSI rises 0. 25 0. 45 0. 35 0. 65 1. 0 Totals n The conditional probability function of X given Y: PX|Y(x | y) = P(X = x | Y = y) if P(Y = y) > 0 PX|Y(x | y) =0 if P(Y = y) = 0 n For each fixed y this is a probability function for X, i. e. the conditional probability function is non-negative and XPX|Y(x | y) = 1. n By the definition of conditional probability, PX|Y(x | y) = PX, Y(x, y)/ PY(y). E. g. , P(HSI rises| Rainy) = 0. 2/0. 35. n When X and Y are independent, PX|Y(x | y) is equal to PX(x). Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 17

Example: Conditional Distributions Y X HSI falls HSI rises Totals P(Y|HSI falls) PX|Y(x | y) = PX, Y(x, y)/ PY(y). Rainy Not Rainy Totals P(X|Rainy) P(X| Not Rainy) 0. 15 0. 4 0. 55 0. 15/0. 35 0. 4/0. 65 0. 25 0. 45 0. 2/0. 35 0. 25/0. 65 0. 35 0. 65 1. 0 0. 15/0. 55 0. 4/0. 55 1. 0 P(Y|HSI 0. 2/0. 45 0. 25/0. 45 rises) PY|X(y | x) = PX, Y(x, y)/ PX(x). Ka-fu Wong © 2004 1. 0 ECON 1003: Analysis of Economic Data 18

Transformation of Random variables n A transformation of random variable(s) results in a new random variable. n For example, if X and Y are random variables, the following are also random variables: n Z=2 X n Z=3+2 X n Z=X 2 n Z=log(X) n Z=X+Y n Z=X 2+Y 2 Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 19

The Expectation (mean) of a Discrete Probability Distribution n The expectation (mean): n reports the central location of the data. n is the long-run average value of the random variable. That is, the average of the outcomes of many experiments. n is also referred to as its expected value, E(X), in a probability distribution. n Is also known as first moment of a random variable. n is a weighted average. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 20

Moments of a random variable The n-th moment is defined as the expectation of the n-th power of a random variable: E(Xn) E(X) First moment E(X 2) Second moment The n-th centralized moment is defined as: E[X-E(X)]n E(X- )2 Second centralized moment E(X- )3 Third centralized moment Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 21

The Expectation (Mean) of Discrete Probability Distribution n For univariate probability distribution, the expectation or mean E(X) is computed by the formula: n For bivariate probability distribution, the expectation or mean E(X) is computed by the formula: Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 22

Conditional Mean of Bivariate Discrete Probability Distribution n For bivariate probability distribution, the conditional expectation or conditional mean E(X|Y) is computed by the formula: n Unconditional expectation or mean of X, E(X) Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 23

Expectation of a linear transformed random variable n If a and b are constants and X is a random variable, then E(a) = a E(b. X) = b. E(X) E(a+b. X) = a+b. E(X) Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 24

The Variance of a Discrete Probability Distribution n The variance measures the amount of spread (variation) of a distribution. n The variance of a discrete distribution is denoted by the Greek letter 2 (sigma squared). n The standard deviation is the square root of 2. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 25

The Variance of a Discrete Probability Distribution n For univariate discrete probability distribution n For bivariate discrete probability distribution Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 26

Variance of a linear transformed random variable n If a and b are constants and X is a random variable, then V(a) = 0 V(b. X) = b 2 V(X) V(a+b. X) = b 2 V(X) Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 27

The Covariance of a Bivariate Discrete Probability Distribution Covariance measures how two random variables

Covariance of linear transformed random variables n If a and b are constants and X is a random variable, then C(a, b) = 0 C(a, b. X) = 0 C(a+b. X, Y) = b. C(X, Y) Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 29

Variance of a sum of random variables n If a and b are constants and X and Y are random variables, then V(X+Y) = V(X) + V(Y) + 2 C(X, Y) V(a. X+b. Y) =a 2 V(X) + b 2 V(Y) + 2 ab. C(X, Y) Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 30

Correlation coefficient n The strength of the dependence between X and Y is measured

EXAMPLE n Dan Desch, owner of College Painters, studied his records for the past 20 weeks and reports the following number of houses painted per week: Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 32

EXAMPLE n Compute the mean and variance of the number of houses painted per

Binomial Probability Distribution n The binomial distribution has the following characteristics: n An outcome of an experiment is classified into one of two mutually exclusive categories, such as a success or failure. n The data collected are the results of counts in a series of trials. n The probability of success stays the same for each trial. n The trials are independent. n For example, tossing an unfair coin three times. n H is labeled success and T is labeled failure. n The data collected are number of H in the three tosses. n The probability of H stays the same for each toss. n The results of the tosses are independent. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 34

Binomial Probability Distribution n To construct a binomial distribution, let n n be the number of trials n x be the number of observed successes n be the probability of success on each trial n The formula for the binomial probability distribution is: P(x) = n. Cx x(1 - )n-x Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 35

The density functions of binomial distributions with n=20 and different success rates p Ka-fu

EXAMPLE x = number of patients who will experience nausea following treatment with Phe-Mycin n = 4 , p = 0. 1 , q = 1 – p = 1 - 0. 1 = 0. 9 Find the probability that 2 of the 4 patients treated will experience nausea. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 37

Binomial Probability Distribution n The formula for the binomial probability distribution is: P(x) = n. Cx x(1 - )n-x TTT, TTH, THT, THH, HTT, HTH, HHT, HHH. n X=number of heads n The coin is fair, i. e. , P(head) = 1/2. n P(x=0) = 3 C 0 0. 5 0(1 - 0. 5)3 -0 =3!/(0!3!) (1/8)=1/8 n P(x=1) = 3 C 1 0. 5 1(1 - 0. 5)3 -1 =3!/(1!2!) (1/8)= 3/8 n P(x=2) = 3 C 2 0. 5 2(1 - 0. 5)3 -2 =3!/(2!1!) (1/8)= 3/8 n P(x=3) = 3 C 3 0. 5 3(1 - 0. 5)3 -3 =3!/(3!0!) (1/8)= 1/8 When the coin is not fair, simple counting rule will not work. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 38

Mean & Variance of the Binomial Distribution n The mean is found by: n

EXAMPLE The Alabama Department of Labor reports that 20% of the workforce in Mobile is unemployed. From a sample of 14 workers, calculate the following probabilities: n Exactly three are unemployed. n At least one are unemployed. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 40

EXAMPLE n The probability of exactly 3: n The probability of at least 3

EXAMPLE n Since =. 2 and n=14. n Hence, the mean is: = n

Finite Population n A finite population is a population consisting of a fixed number of known individuals, objects, or measurements. Examples include: n The number of students in this class. n The number of cars in the parking lot. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 43

Hypergeometric Distribution n The hypergeometric distribution has the following characteristics: n There are only 2 possible outcomes. n The probability of a success is not the same on each trial. n It results from a count of the number of successes in a fixed number of trials. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 44

EXAMPLE In a bag containing 7 red chips and 5 blue chips you select 2 chips one after the other without replacement. 6/11 7/12 5/12 R 1 5/11 B 2 7/11 R 2 B 1 4/11 B 2 The probability of a success (red chip) is not the same on each trial. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 45

Hypergeometric Distribution n The formula for finding a probability using the hypergeometric distribution is: where N is the size of the population, S is the number of successes in the population, x is the number of successes in a sample of n observations. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 46

Hypergeometric Distribution n Use the hypergeometric distribution to find the probability of a specified number of successes or failures if: n the sample is selected from a finite population without replacement (recall that a criteria for the binomial distribution is that the probability of success remains the same from trial to trial) n the size of the sample n is greater than 5% of the size of the population N. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 47

The density functions of hypergeometric distributions with N=100, n=20 and different success rates p

EXAMPLE: Hypergeometric Distribution n The National Air Safety Board has a list of 10 reported safety violations. Suppose only 4 of the reported violations are actual violations and the Safety Board will only be able to investigate five of the violations. What is the probability that three of five violations randomly selected to be investigated are actually violations? Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 49

Poisson Probability Distribution The formula for the binomial probability distribution is: P(x) = n. Cx x(1 - )n-x n The binomial distribution becomes more skewed to the right (positive) as the probability of success become smaller. n The limiting form of the binomial distribution where the probability of success is small and n is large is called the Poisson probability distribution. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 50

Poisson Probability Distribution The Poisson distribution can be described mathematically using the formula: where is the mean number of successes in a particular interval of time, e is the constant 2. 71828, and x is the number of successes. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 51

Poisson Probability Distribution n The mean number of successes can be determined in binomial situations by n , where n is the number of trials and the probability of a success. n The variance of the Poisson distribution is also equal to n . n X, the number of success generally has no specific upper limit. n Probability distribution always skewed to the right. n Becomes symmetrical when gets large. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 52

EXAMPLE: Poisson Probability Distribution n The Sylvania Urgent Care facility specializes in caring for minor injuries, colds, and flu. For the evening hours of 610 PM the mean number of arrivals is 4. 0 per hour. What is the probability of 2 arrivals in an hour? Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 53

Example: Poisson Probabilities x = number of Cleveland air traffic control errors during one week = 0. 4 (expected number of errors per week) Find the probability that 3 errors will occur in a week. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 54

Mean and Variance of a Poisson Random Variable If x is a Poisson random

Several Poisson Distributions Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 56

What distributions to use? n Poisson considers the number of times an event occurs over an INTERVAL of TIME or SPACE. Note that we are not considering a sample of given number of observations. n Thus, if we are considering a sample of 10 observations and we are asked to compute the probability of having 6 successes, we should not use Poisson. Instead, we should consider Binomial or Hypergeometric. n Hypergeometric consider the number of successes in a sample when the probability of success varies across trials due to “without replacement” sampling strategy. To compute the Hypergeometric probability, one will need to know N and S separately. n Suppose we know that the probability of success is 0. 3. We are considering a sample of 10 observations and we are asked to compute the probability of having 6 successes. We cannot use Hypergeometric because we do not have N and S separately. Instead, we have to use Binomial. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 57

What distributions to use? Example In a shipment of 15 hard disks, 5 are defective. If 4 of the disks are inspected, what is the probability that exactly 1 is defective? n First, we recognize that it is not Poisson because "4 of the disks are inspected" (i. e. , sample size =4). n Second, it is sampling without replacement because if we were to inspect four disks for defects, we will not want to sample with replacement. n Third, both N (15 hard disks) and S (5 are defective) are given. Hence we will use Hypergeometric. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 58

What distributions to use? Example From an inventory of 48 cars being shipped to local automobile dealers, 12 have had defective radios installed. What is the probability that one particular dealership receiving 8 cars obtains all with defective radios? n First, we recognize that it is not Poisson because 8 cars are “inspected" (i. e. , sample size =8). n Second, it is sampling without replacement because if we were to inspect all 8 cars for defects, we will not want to sample with replacement. n Third, both N (48 cars) and S (12 have defective radio) are given. Hence we will use Hypergeometric. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 59

What distributions to use? Example The number of claims for missing baggage for a well-known airline in a small city averages nine per day. What is the probability that, on a given day, there will be fewer than three claims made? n First, we recognize that it is likely Poisson because “on a given day”. n Second, we are asked to compute the probability of the number of claims larger than some number. There is no limit on the number of claims that can arrive in a given day. n Third, “average per day” is given. Hence we will use Poisson. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 60

What distributions to use? Example When a customer places an order with Rudy’s on-Line Office Supplies, a computerized accounting information system (AIS) automatically checks to see if the customer has exceeded his or her credit limit. Past records indicate that the probability of customers exceeding their credit limit is 0. 05. Suppose that, on a given day, 20 customers place orders. What is the probability that zero customers will exceed their limits? n First, we recognize that it is not Poisson because 20 customers place orders (i. e. , sample size =20). n Second, the probability of drawing a particular type of customers appears the same across trials because “the probability of customers exceeding their credit limit is 0. 05”. n Hence we will use Binomial. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 61