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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-1 Chapter 5 Some Important Discrete Probability Distributions Statistics.

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Presentation on theme: "Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-1 Chapter 5 Some Important Discrete Probability Distributions Statistics."— Presentation transcript:

1 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-1 Chapter 5 Some Important Discrete Probability Distributions Statistics for Managers Using Microsoft ® Excel 4 th Edition

2 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-2 Chapter Goals After completing this chapter, you should be able to:  Interpret the mean and standard deviation for a discrete probability distribution  Explain covariance and its application in finance  Use the binomial probability distribution to find probabilities  Describe when to apply the binomial distribution  Use the hypergeometric and Poisson discrete probability distributions to find probabilities

3 Statistics for Managers Using Microsoft Excel, 4e © 2004 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

4 Statistics for Managers Using Microsoft Excel, 4e © 2004 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)

5 Statistics for Managers Using Microsoft Excel, 4e © 2004 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 X X Value Probability 0 1/4 = /4 = /4 = Probability

6 Statistics for Managers Using Microsoft Excel, 4e © 2004 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.25) + (1 x.50) + (2 x.25) = 1.0 X P(X)

7 Statistics for Managers Using Microsoft Excel, 4e © 2004 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)

8 Statistics for Managers Using Microsoft Excel, 4e © 2004 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

9 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-9 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 condition affecting the i th outcome of X and the i th outcome of Y

10 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-10 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.2 Recession- $ 25 - $200.5 Stable Economy Expanding Economy Investment E(X) = μ X = (-25)(.2) +(50)(.5) + (100)(.3) = 50 E(Y) = μ Y = (-200)(.2) +(60)(.5) + (350)(.3) = 95

11 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-11 Computing the Standard Deviation for Investment Returns P(X i Y i ) Economic condition Passive Fund X Aggressive Fund Y.2 Recession- $ 25 - $200.5 Stable Economy Expanding Economy Investment

12 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-12 Computing the Covariance for Investment Returns P(X i Y i ) Economic condition Passive Fund X Aggressive Fund Y.2 Recession- $ 25 - $200.5 Stable Economy Expanding Economy Investment

13 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-13 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

14 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-14 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:

15 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-15 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

16 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-16 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

17 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-17 Probability Distributions Continuous Probability Distributions Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions Normal Uniform Exponential Ch. 5Ch. 6

18 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-18 The Binomial Distribution Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions

19 Statistics for Managers Using Microsoft Excel, 4e © 2004 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

20 Statistics for Managers Using Microsoft Excel, 4e © 2004 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

21 Statistics for Managers Using Microsoft Excel, 4e © 2004 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

22 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-22 Rule of Combinations  The number of combinations of selecting X objects out of n objects is where: n! =n(n - 1)(n - 2)... (2)(1) X! = X(X - 1)(X - 2)... (2)(1) 0! = 1 (by definition)

23 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-23 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 = p = (1 -.5) =.5 X = 0, 1, 2, 3, 4 Binomial Distribution Formula

24 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-24 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 =.1

25 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-25 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 =.1  Here, n = 5 and p =.5

26 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-26 Binomial Distribution Characteristics  Mean  Variance and Standard Deviation Wheren = sample size p = probability of success (1 – p) = probability of failure

27 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-27 n = 5 p = 0.1 n = 5 p = 0.5 Mean X P(X) X P(X) 0 Binomial Characteristics Examples

28 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-28 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 =.35, x = 3: P(x = 3|n =10, p =.35) =.2522 n = 10, p =.75, x = 2: P(x = 2|n =10, p =.75) =.0004

29 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-29 Using PHStat  Select PHStat / Probability & Prob. Distributions / Binomial…

30 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-30 Using PHStat  Enter desired values in dialog box Here:n = 10 p =.35 Output for X = 0 to X = 10 will be generated by PHStat Optional check boxes for additional output (continued)

31 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-31 P(X = 3 | n = 10, p =.35) =.2522 PHStat Output P(X > 5 | n = 10, p =.35) =.0949

32 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-32 The Hypergeometric Distribution Binomial Poisson Probability Distributions Discrete Probability Distributions Hypergeometric

33 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-33 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

34 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-34 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

35 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-35 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

36 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-36 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%.

37 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-37 Hypergeometric Distribution in PHStat  Select: PHStat / Probability & Prob. Distributions / Hypergeometric …

38 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-38 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)

39 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-39 The Poisson Distribution Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions

40 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-40 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)

41 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-41 Poisson Distribution Formula where: X = number of successes per unit = expected number of successes per unit e = base of the natural logarithm system ( )

42 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-42 Poisson Distribution Characteristics  Mean  Variance and Standard Deviation where = expected number of successes per unit

43 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-43 Using Poisson Tables X Example: Find P(X = 2) if =.50

44 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-44 Graph of Poisson Probabilities X = P(X = 2) =.0758 Graphically: =.50

45 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-45 Poisson Distribution Shape  The shape of the Poisson Distribution depends on the parameter : = 0.50 = 3.00

46 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-46 Poisson Distribution in PHStat  Select: PHStat / Probability & Prob. Distributions / Poisson…

47 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-47 Poisson Distribution in PHStat  Complete dialog box entries and get output … P(X = 2) = (continued)

48 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-48 Chapter Summary  Addressed the probability of a discrete random variable  Defined covariance and discussed its application in finance  Discussed the Binomial distribution  Discussed the Hypergeometric distribution  Reviewed the Poisson distribution


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