Chapter 5 Discrete Probability Distributions Yandell Econ 216

Chapter 5 Discrete Probability Distributions Yandell – Econ 216 Chap 5 -1

Chapter Goals After completing this chapter, you should be able to: n Distinguish between discrete and continuous probability distributions n Apply the Binomial probability distribution to find probabilities n Use PHStat to find Binomial probabilities Yandell – Econ 216 Chap 5 -2

Introduction to Probability Distributions n Random Variable n Represents a possible numerical value from a random event Random Variables Discrete Random Variable Yandell – Econ 216 Continuous Random Variable Chap 5 -3

Discrete Random Variables n Can only assume a countable number of values Examples: n n Yandell – Econ 216 Roll a die twice Let X be the number of times 4 comes up (then X could be 0, 1, or 2 times) Toss a coin 5 times. Let X be the number of heads (then X = 0, 1, 2, 3, 4, or 5) Chap 5 -4

Discrete Probability Distribution Experiment: Toss 2 Coins. T T H H Yandell – Econ 216 T H Probability Distribution X Value Probability 0 1/4 =. 25 1 2/4 =. 50 2 1/4 =. 25 Probability 4 possible outcomes Let X = # heads. listen . 50. 25 0 1 2 X Chap 5 -5

Discrete Probability Distribution n A list of all possible [ Xi , P(Xi) ] pairs Xi = Value of Random Variable (Outcome) P(Xi) = Probability Associated with Value n n Yandell – Econ 216 Xi’s are mutually exclusive (no overlap) Xi’s are collectively exhaustive (nothing left out) 0 £ P(Xi) £ 1 for each Xi P(Xi) = 1 Chap 5 -6

Discrete Random Variable Summary Measures n Expected Value of a discrete distribution (Weighted Average) E(X) = Xi P(Xi) n Example: Toss 2 coins, X = # of heads, compute expected value of X: X P(X) 0 . 25 1 . 50 2 . 25 E(X) = (0 x. 25) + (1 x. 50) + (2 x. 25) = 1. 0 Yandell – Econ 216 Chap 5 -7

Discrete Random Variable Summary Measures n (continued) Standard Deviation of a discrete distribution where: E(X) = Expected value of the random variable X = Values of the random variable P(X) = Probability of the random variable having the value of X Yandell – Econ 216 Chap 5 -8

Discrete Random Variable Summary Measures n (continued) Example: Toss 2 coins, X = # heads, compute standard deviation (recall E(X) = 1) Possible number of heads = 0, 1, or 2 Yandell – Econ 216 Chap 5 -9

Two Discrete Random Variables n Expected value of the sum of two discrete random variables: E(X + Y) = E(X) + E(Y) = X P(X) + Y P(Y) (The expected value of the sum of two random variables is the sum of the two expected values) Yandell – Econ 216 Chap 5 -10

Covariance n Covariance between two discrete random variables: σXY = [Xi – E(X)][Yj – E(Y)]P(Xi. Yj) where: Xi = possible values of the X discrete random variable Yj = possible values of the Y discrete random variable P(Xi , Yj) = joint probability of the values of Xi and Yj occurring Yandell – Econ 216 Chap 5 -11

Interpreting Covariance n Covariance between two discrete random variables: XY > 0 X and Y tend to move in the same direction XY < 0 X and Y tend to move in opposite directions XY = 0 X and Y do not move closely together Yandell – Econ 216 Chap 5 -12

Probability Distributions Ch. 5 Discrete Probability Distributions Binomial Normal Poisson Uniform Hypergeometric Yandell – Econ 216 Continuous Probability Distributions Ch. 6 Exponential Chap 5 -13

Continuous Probability Distributions n A continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values) n n n thickness of an item time required to complete a task temperature of a solution height, in inches These can potentially take on any value, depending only on the ability to measure accurately Yandell – Econ 216 Chap 5 -14

Discrete Probability Distributions n A discrete random variable is a variable that can assume only a countable number of values Many possible outcomes: n number of complaints per day n number of TV’s in a household n number of rings before the phone is answered Only two possible outcomes: n gender: male or female n defective: yes or no n spreads peanut butter first vs. spreads jelly first Yandell – Econ 216 Chap 5 -15

Application: Return on Investment Let X measure the return on an investment measured in percent and let P(X) be the probability associated with each return. n The list of outcomes and their associated probabilities is given in the table on the next slide. n Yandell – Econ 216 Chap 5 -16

A Discrete Probability Distribution This row means that the probability of a 10% return is 15%. Yandell – Econ 216 X (%) P(X) 9 0. 05 10 0. 15 11 0. 30 12 0. 20 13 0. 15 14 0. 10 15 0. 05 Note that these probabilities sum to one, so this list is exhaustive. Chap 5 -17

Graphically Yandell – Econ 216 Chap 5 -18

Problems The probability of at least a 14% return is: P(X 14) = P(X=14) + P(X=15) = 0. 10 + 0. 05 = 0. 15 Yandell – Econ 216 X (%) P(X) 9 0. 05 10 0. 15 11 0. 30 12 0. 20 13 0. 15 14 0. 10 15 0. 05 Chap 5 -19

A Useful Trick What is the probability of earning less than 14%? The hard way to find this is: P(X < 14) = P(X=9) + P(X=10) + P(X=11) + P(X=12) + P(X=13) The easy way is: P(X < 14) = 1 - P(X 14) = 1. 0 - 0. 15 = 0. 85 Yandell – Econ 216 X (%) P(X) 9 0. 05 10 0. 15 11 0. 30 12 0. 20 13 0. 15 14 0. 10 15 0. 05 Chap 5 -20

The Mean of a discrete distribution is the weighted average of the outcomes: n n Which is: Yandell – Econ 216 Chap 5 -21

Calculating the Mean listen X (%) P(X) X P(X) 9 0. 05 0. 45 10 0. 15 1. 50 11 0. 30 3. 30 12 0. 20 2. 40 13 0. 15 1. 95 14 0. 10 1. 40 15 0. 05 0. 75 Mean = 11. 75 Yandell – Econ 216 Chap 5 -22

The Variance of a discrete distribution is the weighted average of the squared deviations from the mean: n n Which becomes: Yandell – Econ 216 Chap 5 -23

Calculating the Variance X-m (X - m) 2 P(X) 9 - 11. 75 7. 5625 0. 378 10 - 11. 75 3. 0625 0. 459 11 - 11. 75 0. 5625 0. 169 12 - 11. 75 0. 0625 0. 013 13 - 11. 75 1. 5625 0. 234 14 - 11. 75 5. 0625 0. 506 15 - 11. 75 10. 5625 0. 528 Variance = 2. 288 (PHStat will do the calculations for the mean and variance for us nwhen we use the Binomial distribution) n Yandell – Econ 216 Chap 5 -24

The Standard Deviation The standard deviation is the square root of the variance: Empirical Rule: the mean 2 standard deviations includes about 95% of the observations Here, So we know that about 95% of the values from this distribution fall between 8. 75 and 14. 75. Yandell – Econ 216 Chap 5 -25

Mean 2 Standard Deviations +/- 2σ m Yandell – Econ 216 Chap 5 -26

Practice The probability distribution for damage claims paid per year for collision insurance is: n Yandell – Econ 216 X ($) P(X) 0 0. 90 400 0. 04 1000 0. 03 2000 0. 01 4000 0. 01 6000 0. 01 Chap 5 -27

Question 1 1. Yandell – Econ 216 Find the expected collision payment to determine the annual collision insurance premium that will enable the company to break even. Chap 5 -28

Question 2 2. Yandell – Econ 216 The insurance company charges an annual rate of $260 for collision coverage. What is the expected value of the collision coverage to a policyholder? (expected payments from the company minus cost of the coverage) Chap 5 -29

Another type of customer Consider a different class of customer with the following claim distribution: n Yandell – Econ 216 X ($) P(X) 0 0. 80 400 0. 06 1000 0. 05 2000 0. 03 4000 0. 03 6000 0. 03 Chap 5 -30

Questions 3&4 How do your answers change for questions 1&2 using this different type of customer? n n How do these customers compare? Yandell – Econ 216 Chap 5 -31

The Binomial Distribution Probability Distributions Discrete Probability Distributions Binomial Poisson Hypergeometric Yandell – Econ 216 Chap 5 -32

The Binomial Distribution n Characteristics of the Binomial Distribution: n n n Yandell – Econ 216 A trial has only two possible outcomes – “success” or “failure” There is a fixed number, n, of identical trials The trials of the experiment are independent of each other The probability of a success, π, remains constant from trial to trial If π represents the probability of a success, then (1 - π) is the probability of a failure Chap 5 -33

Binomial Distribution Settings n n Yandell – Econ 216 A manufacturing plant labels items as either defective or acceptable A firm bidding for a contract will either get the contract or not A marketing research firm receives survey responses of “yes I will buy” or “no I will not” New job applicants either accept the offer or reject it Chap 5 -34

Counting Rule for Combinations n A combination is an outcome of an experiment where x objects are selected from a group of n objects where: n! =n(n - 1)(n - 2). . . (2)(1) x! = x(x - 1)(x - 2). . . (2)(1) 0! = 1 Yandell – Econ 216 (by definition) Chap 5 -35

Binomial Distribution Formula listen n! x n-x P(x) = π (1 - π) x ! (n - x )! P(X = x) = probability of x successes in n trials, with probability of success π on each trial x = number of ‘successes’ in sample, (x = 0, 1, 2, . . . , n) π = probability of “success” per trial 1 - π = probability of “failure” n = number of trials (sample size) Yandell – Econ 216 Example: Flip a coin four times, let X = # heads: n=4 π = 0. 5 1 - π = (1 -. 5) =. 5 x = 0, 1, 2, 3, 4 Chap 5 -36

Don’t Worry We will use PHStat to get Binomial Probabilities More examples and sample PHStat output appear soon Yandell – Econ 216 Chap 5 -37

Binomial Distribution n The shape of the binomial distribution depends on the values of π and n Mean n Here, n = 5 and π =. 1 . 6. 4. 2 0 P(x) X 0 listen n Here, n = 5 and π =. 5 . 6. 4. 2 0 P(x) 1 2 3 4 5 n = 5, π = 0. 5 X 0 Yandell – Econ 216 n = 5, π = 0. 1 1 2 3 4 5 Chap 5 -38

Binomial Distribution Characteristics n n Mean Variance and Standard Deviation Where n = sample size π = probability of success 1 – π = probability of failure Yandell – Econ 216 Chap 5 -39

Binomial Characteristics Examples Mean . 6. 4. 2 0 P(x) X 0 . 6. 4. 2 0 P(x) 1 2 3 4 5 n = 5, π = 0. 5 X 0 Yandell – Econ 216 n = 5, π = 0. 1 1 2 3 4 5 Chap 5 -40

Using Binomial Tables n x π=. 15 π=. 20 π=. 25 π=. 30 π=. 35 π=. 40 π=. 45 π=. 50 10 0 1 2 3 4 5 6 7 8 9 10 0. 1969 0. 3474 0. 2759 0. 1298 0. 0401 0. 0085 0. 0012 0. 0001 0. 0000 0. 1074 0. 2684 0. 3020 0. 2013 0. 0881 0. 0264 0. 0055 0. 0008 0. 0001 0. 0000 0. 0563 0. 1877 0. 2816 0. 2503 0. 1460 0. 0584 0. 0162 0. 0031 0. 0004 0. 0000 0. 0282 0. 1211 0. 2335 0. 2668 0. 2001 0. 1029 0. 0368 0. 0090 0. 0014 0. 0001 0. 0000 0. 0135 0. 0725 0. 1757 0. 2522 0. 2377 0. 1536 0. 0689 0. 0212 0. 0043 0. 0005 0. 0000 0. 0060 0. 0403 0. 1209 0. 2150 0. 2508 0. 2007 0. 1115 0. 0425 0. 0106 0. 0016 0. 0001 0. 0025 0. 0207 0. 0763 0. 1665 0. 2384 0. 2340 0. 1596 0. 0746 0. 0229 0. 0042 0. 0003 0. 0010 0. 0098 0. 0439 0. 1172 0. 2051 0. 2461 0. 2051 0. 1172 0. 0439 0. 0098 0. 0010 10 9 8 7 6 5 4 3 2 1 0 10 π=. 85 π=. 80 π=. 75 π=. 70 π=. 65 π=. 60 π=. 55 π=. 50 x n Examples: n = 10, π =. 35, x = 3: P(X = 3|n =10, π =. 35) =. 2522 n = 10, π =. 75, x = 2: P(X = 2|n =10, π =. 75) =. 0004 Yandell – Econ 216 Chap 5 -41

Using PHStat In Excel 2007 n Yandell – Econ 216 Select PHStat / Probability & Prob. Distributions / Binomial… Chap 5 -42

In Excel 2010 or 2013 Yandell – Econ 216 Chap 5 -43

Using PHStat (continued) n Enter desired values in dialog box Here: n = 10 π =. 35 Output for X = 0 to X = 10 will be generated by PHStat Optional check boxes for additional output Yandell – Econ 216 Chap 5 -44

PHStat Output P(X = 3 | n = 10, π =. 35) =. 2522 P(X > 5 | n = 10, π =. 35) =. 0949 Yandell – Econ 216 Chap 5 -45

Sample Problem: A Light Bulb Shipment Consider a large shipment of light bulbs. We will select a sample of five bulbs and determine whether to accept or reject the whole shipment. Binomial Assumptions are satisfied: n ‘n’ Identical Trials 5 light bulbs randomly taken from a shipment n 2 Mutually Exclusive Outcomes in Each Trial defective or not defective light bulb n Constant Probability For Each Trial each light bulb has the same probability, π, of being not defective Yandell – Econ 216 Chap 5 -46

The Light Bulb Problem (continued) Suppose that we randomly select five (n = 5) light bulbs from a shipment. We will accept the shipment if at least four bulbs work. If each bulb has a π =. 90 chance of working, what is the probability that we will accept the shipment? n That is, what is P(X 4) ? Yandell – Econ 216 Chap 5 -47

Ph. Stat Demo Click here to view the PHStat demo for the light bulb problem: n Click here to watch the demonstration Yandell – Econ 216 Chap 5 -48

The Poisson Distribution Probability Distributions Discrete Probability Distributions Binomial Poisson Hypergeometric Yandell – Econ 216 Chap 5 -49

The Poisson Distribution n Characteristics of the Poisson Distribution: n n Yandell – Econ 216 The outcomes of interest are relative to the possible outcomes The average number of outcomes of interest per unit is (lambda) The number of outcomes of interest are random, and the occurrence of one outcome does not influence the chances of another outcome of interest The probability of that an outcome of interest occurs in a given segment is the same for all segments Chap 5 -50

Poisson Distribution Formula where: x = number of successes per unit = expected number of successes e = base of the natural logarithm system (2. 71828. . . ) Yandell – Econ 216 Chap 5 -51

Poisson Distribution Characteristics n n Mean Variance and Standard Deviation where Yandell – Econ 216 = expected number of successes per unit Chap 5 -52

Using Poisson Tables X 0. 10 0. 20 0. 30 0. 40 0. 50 0. 60 0. 70 0. 80 0. 90 0 1 2 3 4 5 6 7 0. 9048 0. 0905 0. 0045 0. 0002 0. 0000 0. 8187 0. 1637 0. 0164 0. 0011 0. 0000 0. 7408 0. 2222 0. 0333 0. 0003 0. 0000 0. 6703 0. 2681 0. 0536 0. 0072 0. 0007 0. 0001 0. 0000 0. 6065 0. 3033 0. 0758 0. 0126 0. 0016 0. 0002 0. 0000 0. 5488 0. 3293 0. 0988 0. 0198 0. 0030 0. 0004 0. 0000 0. 4966 0. 3476 0. 1217 0. 0284 0. 0050 0. 0007 0. 0001 0. 0000 0. 4493 0. 3595 0. 1438 0. 0383 0. 0077 0. 0012 0. 0000 0. 4066 0. 3659 0. 1647 0. 0494 0. 0111 0. 0020 0. 0003 0. 0000 Example: Find P(X = 2) if =. 50 Yandell – Econ 216 Chap 5 -53

Graph of Poisson Probabilities Graphically: =. 50 X = 0. 50 0 1 2 3 4 5 6 7 0. 6065 0. 3033 0. 0758 0. 0126 0. 0016 0. 0002 0. 0000 Yandell – Econ 216 P(X = 2) =. 0758 Chap 5 -54

Poisson Distribution Shape n The shape of the Poisson Distribution depends on the parameter : = 0. 50 Yandell – Econ 216 = 3. 00 Chap 5 -55

The Hypergeometric Distribution Probability Distributions Discrete Probability Distributions Binomial Poisson Hypergeometric Yandell – Econ 216 Chap 5 -56

The Hypergeometric Distribution n “n” trials in a sample taken from a finite population of size N n Sample taken without replacement n Trials are dependent n Concerned with finding the probability of “X” successes in the sample where there are “A” successes in the population Yandell – Econ 216 Chap 5 -57

Hypergeometric Distribution Formula (Two possible outcomes per trial) Where Yandell – Econ 216 N = Population size A = number of successes in the population N – A = number of failures in the population n = sample size X = number of successes in the sample n – X = number of failures in the sample Chap 5 -58

Hypergeometric Distribution Formula ■ Example: 3 Light bulbs were selected from 10. Of the 10 there were 4 defective. What is the probability that 2 of the 3 selected are defective? N = 10 A=4 Yandell – Econ 216 n=3 X=2 Chap 5 -59

Hypergeometric Distribution in PHStat n Select: Add Ins / PHStat / Probability & Prob. Distributions / Hypergeometric … Yandell – Econ 216 Chap 5 -60

Hypergeometric Distribution in PHStat (continued) Complete dialog box entries and get output … n N = 10 A=4 n=3 X=2 P(X = 2) = 0. 3 Yandell – Econ 216 Chap 5 -61

Chapter Summary n n Distinguished between discrete and continuous probability distributions Examined discrete probability distributions and their summary measures (Binomial, Poisson, and Hypergeometric) Yandell – Econ 216 Chap 5 -62