 # Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-1 Chapter 5 Some Important Discrete Probability Distributions Statistics for.

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Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-1 Chapter 5 Some Important Discrete Probability Distributions Statistics for Managers 4 th Edition

Business Statistics, A First Course (4e) © 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, variance, and standard deviation of a probability distribution  To calculate probabilities from Binomial,  Hypergeometric and Poisson distributions  How to use the Binomial, Hypergeometric and Poisson distributions to solve business problems

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-3 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

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-4 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)

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-5 Experiment: Toss 2 Coins. Let X = # heads. T T Discrete Probability Distribution 4 possible outcomes T T H H HH Probability Distribution 0 1 2 X X Value Probability 0 1/4 = 0.25 1 2/4 = 0.50 2 1/4 = 0.25 0.50 0.25 Probability

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-6 Discrete Random Variable Summary Measures  Expected Value (or mean) of a discrete distribution (Weighted Average)  Example: Toss 2 coins, X = # of heads, compute expected value of X: E(X) = (0 x 0.25) + (1 x 0.50) + (2 x 0.25) = 1.0 X P(X) 0 0.25 1 0.50 2 0.25

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-7  Variance of a discrete random variable  Standard Deviation of a discrete random variable where: E(X) = 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)

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-8  Example: Toss 2 coins, X = # heads, compute standard deviation (recall E(X) = 1) Discrete Random Variable Summary Measures (continued) Possible number of heads = 0, 1, or 2

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-9 Example of Expected Value A company is considering a proposal to develop a new product. The initial cash outlay would be \$1 million, and development time would be three years. If successful, the firm anticipates that net profit (revenue minus initial cash outlay) over the 5 year life cycle of the product will be \$1.5 million. If moderately successful, net profit will reach \$1.2 million. If unsuccessful, the firm anticipates zero cash inflows. The firms assigns the following probabilities to the 5- year prospects for this product: successful,.60; moderately successful,.30; and unsuccessful,.10. What is the expected net profit? Ignore time value of money.

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-10 Calculation of Expected Value To calculate the expected value E(X) = Σ [(X) P(X)] = (1.5)(.6) + (1.2)(.3) + (-1)(.1) = + 1.16 Since the expected value Is positive this project passes the initial approval process

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 5-11 Investment Returns The Mean Consider the return per \$1000 for two types of investments. Economic P(X i Y i ) Condition Investment Passive Fund XAggressive Fund Y 0.2 Recession - \$25- \$200 0.5 Stable Economy + \$50 + \$60 0.3 Expanding Economy+ \$100+ \$350

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice- Hall, Inc. Chap 5-12 Investment Returns The Mean E(X) = μ X = (-25)(.2) +(50)(.5) + (100)(.3) = 50 E(Y) = μ Y = (-200)(.2) +(60)(.5) + (350)(.3) = 95 Interpretation: Fund X is averaging a \$50.00 return and fund Y is averaging a \$95.00 return per \$1000 invested.

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice- Hall, Inc. Chap 5-13 Investment Returns Standard Deviation Interpretation: Even though fund Y has a higher average return, it is subject to much more variability and the probability of loss is higher.

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice- Hall, Inc. Chap 5-14 Investment Returns Covariance Interpretation: Since the covariance is large and positive, there is a positive relationship between the two investment funds, meaning that they will likely rise and fall together.

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice- Hall, Inc. Chap 5-15 The Sum of Two Random Variables: Measures  Expected Value:  Variance:  Standard deviation:

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice- Hall, Inc. Chap 5-16 Portfolio Expected Return and Expected Risk  Investment portfolios usually contain several different funds (random variables)  The expected return and standard deviation of two funds together can now be calculated.  Investment Objective: Maximize return (mean) while minimizing risk (standard deviation).

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice- Hall, Inc. Chap 5-17 Portfolio Expected Return and Expected 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

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice- Hall, Inc. Chap 5-18 Portfolio Expect Return and Expected Risk Recall: Investment X: E(X) = 50 σ X = 43.30 Investment Y: E(Y) = 95 σ Y = 193.21 σ XY = 8250 Suppose 40% of the portfolio is in Investment X and 60% is in Investment Y:  The portfolio return is between the values for investments X and Y considered individually.

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-19 Binomial Probability Distribution  A fixed number of observations, n  e.g., 15 tosses of a coin; ten light bulbs taken from a warehouse  Two mutually exclusive and collectively exhaustive categories  e.g., head or tail in each toss of a coin; defective or not defective light bulb  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

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-20 Binomial Probability Distribution (continued)  Observations are independent  The outcome of one observation does not affect the outcome of the other  Two sampling methods  Infinite population without replacement  Finite population with replacement

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-21 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 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

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-22 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) 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 = 0.5 1 - p = (1 - 0.5) = 0.5 X = 0, 1, 2, 3, 4 Binomial Distribution Formula

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-23 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

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-24 Binomial Distribution Characteristics  Mean  Variance and Standard Deviation Wheren = sample size p = probability of success (1 – p) = probability of failure

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-25 Binomial Example A doctor prescribes Nexium to 60% of his patients who have gastro-intestinal problems. What is the probability that out of the next 10 patients: 1. Six of them are given a prescription of Nexium 2. At least 6 are given a prescription of Nexium?

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-26

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-27 The Poisson Distribution  It is appropriate to use the Poisson distribution when:  You have an event that occurs randomly through time and space  You know the average number of successes you expect to observe over a given time frame or in a given space

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-28 Poisson Distribution Formula where: X = number of events in an area of opportunity = expected number of events e = base of the natural logarithm system (2.71828...)

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-29 Poisson Example On average there are three thread defects in a 10 yard bolt of fine wool fabric. What is the probability of finding no more than two thread defects in a randomly chosen 10 bolt lot?

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-30 Solution

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-31 Poisson Distribution Characteristics  Mean  Variance and Standard Deviation where = expected number of events

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-32 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

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-33 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

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-34 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

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-35 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.30, or 30%.

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-36 Hypergeometric Distribution in PHStat  Complete dialog box entries and get output … N = 10 n = 3 A = 4 X = 2 P(X = 2) = 0.3 (continued)

Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-37 Chapter Summary  Addressed the probability of a discrete random variable  Discussed the Binomial distribution  Discussed the Poisson distribution  Discussed the Hypergeometric distribution

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