Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.. Chap 5-1 Chapter 5 Some Important Discrete Probability Distributions Basic Business Statistics

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-2 Learning Objectives In this chapter, you learn:  The properties of a probability distribution  To calculate the expected value and variance of a probability distribution  To calculate the covariance and its use in finance  To calculate probabilities from binomial, hypergeometric, and Poisson distributions  How to use the binomial, hypergeometric, and Poisson distributions to solve business problems

 An experiment is any process that generates an outcome.  For a random experiment, the outcomes cannot be predicted (i.e. they’re uncertain) random variable  A random variable is a variable whose numerical value depends upon the outcome of a random experiment. Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-3 Random Variable

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-4 Random Variable  Random Variable  Outcomes of an experiment expressed numerically  E.g., Toss a die twice; count the number of times the number 4 appears (0, 1 or 2 times)  E.g., Toss a coin; assign $10 to head and -$30 to a tail = $10 T = -$30

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-5 Introduction to Probability Distributions  Random Variable  Represents a possible numerical value from an uncertain event Random Variables Discrete Random Variable Continuous Random Variable Ch. 5Ch. 6

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-6 Discrete Random Variables  Can only assume a countable number of values Examples:  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)

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-7 Discrete Probability Distribution  List of All Possible [X j, P(X j ) ] Pairs  X j = Value of random variable  P(X j ) = Probability associated with value  Mutually Exclusive (Nothing in Common)  Collective Exhaustive (Nothing Left Out)

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-8 Experiment: Toss 2 Coins. Let X = # heads. T T Discrete Probability Distribution 4 possible outcomes T T H H HH Probability Distribution X X Value Probability 0 1/4 = /4 = /4 = Probability

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-9 Discrete Random Variable Summary Measures  Expected Value (or mean) of a discrete distribution (Weighted Average)  Example: Toss 2 coins, X = # of heads, compute the mean of X: μ = (0 x 0.25) + (1 x 0.50) + (2 x 0.25) = 1.0 X P(X)

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-10  Variance of a discrete random variable  Standard Deviation of a discrete random variable where: μ = The mean (Expected value) of the discrete random variable X X i = the i th outcome of X P(X i ) = Probability of the i th occurrence of X Discrete Random Variable Summary Measures (continued)

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-11  Example: Toss 2 coins, X = # heads, compute standard deviation (recall μ = 1) Discrete Random Variable Summary Measures (continued) Possible number of heads = 0, 1, or 2

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-12 Probability Distributions Continuous Probability Distributions Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions Normal Uniform Exponential Ch. 5Ch. 6

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-13 The Binomial Distribution Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-14 Binomial Probability Distribution  A fixed number of observations, n  e.g., 15 tosses of a coin; ten light bulbs taken from a warehouse  All outcomes may be classified as one of two mutually exclusive categories  e.g., head or tail in each toss of a coin; defective or not defective light bulb  Generally called “success” and “failure  Generally called “success” and “failure”  Probability of success is p, probability of failure is 1 – p  Constant probability for each observation  e.g., Probability of getting a tail is the same each time we toss the coin

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-15 Possible Binomial Distribution Settings  A manufacturing plant labels items as either defective or acceptable  A firm bidding for contracts will either get a contract or not  A student taking a multiple choice/true-false  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

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-16 P(X) = probability of X successes in n trials, with probability of success p on each trial X = number of ‘successes’ in sample, (X = 0, 1, 2,..., n) where n = sample size (number of trials or observations) p = probability of “success” P(X) n X ! nX p(1-p) X n X ! ()!    Example: Flip a coin four times, let x = # heads: n = 4 p = p = ( ) = 0.5 X = 0, 1, 2, 3, 4 Binomial Distribution Formula

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-17 Example: Calculating a Binomial Probability What is the probability of one success in five observations if the probability of success is.1? X = 1, n = 5, and p = 0.1

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. The Binomial Distribution, An Example Worked by Equation A study by the International Coffee Association found that 52% of the U.S. population aged 10 and over drink coffee. For a randomly selected group of 4 individuals, what is the probability that 3 of them are coffee drinkers? So, p = 0.52, (1 – p) = 0.48, x = 3, (n – x) = 1.

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. The Binomial Distribution, Working with the Equation To solve the problem, we substitute:

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-20 n = 5 p = 0.1 n = 5 p = 0.5 Mean X P(X) X P(X) 0 Binomial Distribution  The shape of the binomial distribution depends on the values of p and n  Here, n = 5 and p = 0.1  Here, n = 5 and p = 0.5

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-21 Binomial Distribution Characteristics  Mean  Variance and Standard Deviation Wheren = sample size p = probability of success (1 – p) = probability of failure

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-22 n = 5 p = 0.1 n = 5 p = 0.5 Mean X P(X) X P(X) 0 Binomial Characteristics Examples

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-23 Binomial Probability Distribution  A fixed number of observations, n  e.g., 15 tosses of a coin; ten light bulbs taken from a warehouse  All outcomes may be classified as one of two mutually exclusive categories  e.g., head or tail in each toss of a coin; defective or not defective light bulb  Generally called “success” and “failure  Generally called “success” and “failure”  Probability of success is p, probability of failure is 1 – p  Constant probability for each observation  e.g., Probability of getting a tail is the same each time we toss the coin

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-24 P(X) = probability of X successes in n trials, with probability of success p on each trial X = number of ‘successes’ in sample, (X = 0, 1, 2,..., n) where n = sample size (number of trials or observations) p = probability of “success” P(X) n X ! nX p(1-p) X n X ! ()!    Example: Flip a coin four times, let x = # heads: n = 4 p = p = ( ) = 0.5 X = 0, 1, 2, 3, 4 Binomial Distribution Formula

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-25 Using Binomial Tables n = 10 x…p=.20p=.25p=.30p=.35p=.40p=.45p= ………………………………………………………… …p=.80p=.75p=.70p=.65p=.60p=.55p=.50x Examples: n = 10, p = 0.35, x = 3: P(x = 3|n =10, p = 0.35) = n = 10, p = 0.75, x = 2: P(x = 2|n =10, p = 0.75) = Note: The table only goes up to p=.50

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-26 Binomial Distribution in PHStat  Steps: PHStat | Probability & Prob. Distributions | Binomial  Example in Excel Spreadsheet  Or use Table E.6 on pages

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. The Binomial Distribution, An Example Worked by Equation A study by the International Coffee Association found that 52% of the U.S. population aged 10 and over drink coffee. For a randomly selected group of 4 individuals, what is the probability that 3 of them are coffee drinkers? So, p = 0.52, (1 – p) = 0.48, x = 3, (n – x) = 1.

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-28 Example: Binomial Distribution A mid-term exam has 30 multiple choice questions, each with 5 possible answers. What is the probability of randomly guessing the answer for each question and passing the exam (i.e., having guessed at least 18 questions correctly)? Using results from PHStat:

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-29 The Poisson Distribution Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-30 Important Discrete Probability Distributions Discrete Probability Distributions BinomialHypergeometric Poisson Siméon Poisson

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-31 Poisson Distribution Formula where: X = number of events in an area of opportunity = expected number of events e = base of the natural logarithm system ( )

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-32 Poisson Distribution Characteristics  Mean  Variance and Standard Deviation where = expected number of events

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. The Poisson Distribution The Poisson distribution defines the probability that an event will occur exactly x times over a given span of time, space, or distance.  where = the mean number of occurrences over the span e = , a constant  Mean = Variance =  Example: During the 12 p.m. – 1 p.m. noon hour, arrivals at a curbside banking machine have been found to be Poisson distributed with a mean of 1.3 persons per minute. If x = number of arrivals per minute, determine the probability that less than two persons arrive at the banking machine during the next minute: = 1.3P(X < 2) = P(X  1)

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-34 The Poisson Distribution  Apply the Poisson Distribution when:  You wish to count the number of times an event occurs in a given area of opportunity  The probability that an event occurs in one area of opportunity is the same for all areas of opportunity  The number of events that occur in one area of opportunity is independent of the number of events that occur in the other areas of opportunity  The probability that two or more events occur in an area of opportunity approaches zero as the area of opportunity becomes smaller  The average number of events per unit is (lambda)

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. The Poisson Distribution, Working with the Equation Example: During the 12 p.m. – 1 p.m. noon hour, arrivals at a curbside banking machine have been found to be Poisson distributed with a mean of 1.3 persons per minute. If x = number of arrivals per minute, determine P(X < 2):  P(x  2) = =

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-36 Using Poisson Tables X Example: Find P(X = 2) if = 0.50

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-37 Graph of Poisson Probabilities X = P(X = 2) = Graphically: = 0.50

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-38 Poisson Distribution Shape  The shape of the Poisson Distribution depends on the parameter : = 0.50 = 3.00

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-39 Poisson Distribution in PHStat  Steps: PHStat | Probability & Prob. Distributions | Poisson  Example in Excel Spreadsheet  Table E.7 on pp PXx x x (| !  e -

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-40 The Covariance  The covariance measures the strength of the linear relationship between two variables  The covariance: where:X = discrete variable X X i = the i th outcome of X Y = discrete variable Y Y i = the i th outcome of Y P(X i Y i ) = probability of occurrence of the i th outcome of X and the i th outcome of Y

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-41 Computing the Mean for Investment Returns Return per $1,000 for two types of investments P(X i Y i ) Economic condition Passive Fund X Aggressive Fund Y 0.2 Recession- $ 25 - $ Stable Economy Expanding Economy Investment E(X) = μ X = (-25)(0.2) +(50)(0.5) + (100)(0.3) = 50 E(Y) = μ Y = (-200)(0.2) +(60)(0.5) + (350)(0.3) = 95

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-42 Computing the Standard Deviation for Investment Returns P(X i Y i ) Economic condition Passive Fund X Aggressive Fund Y 0.2 Recession- $ 25 - $ Stable Economy Expanding Economy Investment

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-43 Computing the Covariance for Investment Returns P(X i Y i ) Economic condition Passive Fund X Aggressive Fund Y 0.2 Recession- $ 25 - $ Stable Economy Expanding Economy Investment

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-44 Interpreting the Results for Investment Returns  The aggressive fund has a higher expected return, but much more risk μ Y = 95 > μ X = 50 but σ Y = > σ X =  The Covariance of 8250 indicates that the two investments are positively related and will vary in the same direction

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-45 The Hypergeometric Distribution Binomial Poisson Probability Distributions Discrete Probability Distributions Hypergeometric Skip!!

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-46 The Hypergeometric Distribution  “n” trials in a sample taken from a finite population of size N  Sample taken without replacement  Outcomes of trials are dependent  Concerned with finding the probability of “X” successes in the sample where there are “A” successes in the population

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-47 Hypergeometric Distribution Formula Where 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

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-48 Properties of the Hypergeometric Distribution  The mean of the hypergeometric distribution is  The standard deviation is Where is called the “Finite Population Correction Factor” from sampling without replacement from a finite population

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-49 Using the Hypergeometric Distribution ■Example: 3 different computers are checked from 10 in the department. 4 of the 10 computers have illegal software loaded. What is the probability that 2 of the 3 selected computers have illegal software loaded? N = 10n = 3 A = 4 X = 2 The probability that 2 of the 3 selected computers have illegal software loaded is 0.30, or 30%.

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-50 The Sum of Two Random Variables  Expected Value of the sum of two random variables:  Variance of the sum of two random variables:  Standard deviation of the sum of two random variables:

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-51 Portfolio Expected Return and Portfolio Risk  Portfolio expected return (weighted average return):  Portfolio risk (weighted variability) Where w = portion of portfolio value in asset X (1 - w) = portion of portfolio value in asset Y

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-52 Portfolio Example Investment X: μ X = 50 σ X = Investment Y: μ Y = 95 σ Y = σ XY = 8250 Suppose 40% of the portfolio is in Investment X and 60% is in Investment Y: The portfolio return and portfolio variability are between the values for investments X and Y considered individually

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-53 Chapter Summary  Addressed the probability of a discrete random variable  Defined covariance and discussed its application in finance  Discussed the Binomial distribution  Discussed the Poisson distribution  Discussed the Hypergeometric distribution