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© 2002 Prentice-Hall, Inc.Chap 5-1 Basic Business Statistics (8 th Edition) Chapter 5 Some Important Discrete Probability Distributions

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© 2002 Prentice-Hall, Inc. Chap 5-2 Chapter Topics The probability of a discrete random variable Covariance and its applications in finance Binomial distribution Poisson distribution Hypergeometric distribution

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© 2002 Prentice-Hall, Inc. Chap 5-3 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)

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© 2002 Prentice-Hall, Inc. Chap 5-4 Discrete Random Variable Discrete random variable Obtained by counting (1, 2, 3, etc.) Usually a finite number of different values e.g.: Toss a coin five times; Count the number of tails (0, 1, 2, 3, 4, or 5 times)

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© 2002 Prentice-Hall, Inc. Chap 5-5 Probability Distribution Values Probability 01/4 =.25 12/4 =.50 21/4 =.25 Discrete Probability Distribution Example Event: Toss 2 Coins. Count # Tails. T T TT

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© 2002 Prentice-Hall, Inc. Chap 5-6 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) Collectively exhaustive (nothing left out)

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© 2002 Prentice-Hall, Inc. Chap 5-7 Summary Measures Expected value (the mean) Weighted average of the probability distribution:

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© 2002 Prentice-Hall, Inc. Chap 5-8 Summary Measures Example of expected value (the mean): Toss two coins, count the number of tails, compute expected value continued

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© 2002 Prentice-Hall, Inc. Chap 5-9 Summary Measures Variance Weight average squared deviation about the mean (continued)

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© 2002 Prentice-Hall, Inc. Chap 5-10 Example of variance: Toss two coins, count number of tails, compute variance Summary Measures (continued)

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© 2002 Prentice-Hall, Inc. Chap 5-11 Covariance and its Application

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© 2002 Prentice-Hall, Inc. Chap 5-12 Computing the Mean for Investment Returns Return per $1,000 for two types of investments P(X i Y i ) Economic condition Dow Jones fund X Growth Stock Y.2 Recession-$100 -$200.5 Stable Economy+ 100 + 50.3 Expanding Economy + 250 + 350 Investment

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© 2002 Prentice-Hall, Inc. Chap 5-13 Computing the Variance for Investment Returns P(X i Y i ) Economic condition Dow Jones fund X Growth Stock Y.2 Recession-$100 -$200.5 Stable Economy+ 100 + 50.3 Expanding Economy + 250 + 350 Investment

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© 2002 Prentice-Hall, Inc. Chap 5-14 Computing the Covariance for Investment Returns P(X i Y i ) Economic condition Dow Jones fund X Growth Stock Y.2 Recession-$100 -$200.5 Stable Economy+ 100 + 50.3 Expanding Economy + 250 + 350 Investment The Covariance of 23,000 indicates that the two investments are positively related and will vary together in the same direction.

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© 2002 Prentice-Hall, Inc. Chap 5-15 Important Discrete Probability Distributions Discrete Probability Distributions BinomialHypergeometricPoisson

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© 2002 Prentice-Hall, Inc. Chap 5-16 Binomial Probability Distribution “n” Identical trials e.g.: 15 tosses of a coin; 10 light bulbs taken from a warehouse Two mutually exclusive outcomes on each trial e.g.: Heads or tails in each toss of a coin; defective or not defective light bulb Trials are independent The outcome of one trial does not affect the outcome of the other

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© 2002 Prentice-Hall, Inc. Chap 5-17 Binomial Probability Distribution Constant probability for each trial e.g.: Probability of getting a tail is the same each time a coin is tossed Two sampling methods Infinite population without replacement Finite population with replacement (continued)

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© 2002 Prentice-Hall, Inc. Chap 5-18 Binomial Probability Distribution Function Tails in 2 Tosses of Coin X P(X) 0 1/4 =.25 1 2/4 =.50 2 1/4 =.25

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© 2002 Prentice-Hall, Inc. Chap 5-19 Binomial Distribution Characteristics Mean e.g.: Variance and standard deviation e.g.: n = 5 p = 0.1 0.2.4.6 012345 X P(X)

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© 2002 Prentice-Hall, Inc. Chap 5-20 Binomial Distribution in PHStat PHStat | Probability & Prob. Distributions | Binomial Example in Excel Spreadsheet

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© 2002 Prentice-Hall, Inc. Chap 5-21 Poisson Distribution Poisson process Discrete events in an “interval” The probability of one success in an interval is stable The probability of more than one success in this interval is 0 The probability of success is independent from interval to interval e.g.: The number of customers arriving in 15 minutes e.g.: The number of defects per case of light bulbs PXx x x (| ! e -

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© 2002 Prentice-Hall, Inc. Chap 5-22 Poisson Probability Distribution Function e.g.: Find the probability of four customers arriving in three minutes when the mean is 3.6.

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© 2002 Prentice-Hall, Inc. Chap 5-23 Poisson Distribution in PHStat PHStat | probability & probability distributions | Poisson Example in excel spreadsheet

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© 2002 Prentice-Hall, Inc. Chap 5-24 Poisson Distribution Characteristics Mean Standard deviation and variance = 0.5 = 6 0.2.4.6 012345 X P(X) 0.2.4.6 0246810 X P(X)

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© 2002 Prentice-Hall, Inc. Chap 5-25 Hypergeometric Distribution “n” Trials in a sample taken from a finite population of size N Sample taken without replacement Trials are dependent Concerned with finding the probability of “X” successes in the sample where there are “A” successes in the population

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© 2002 Prentice-Hall, Inc. Chap 5-26 Hypergeometric Distribution Function e.g.: Three Light bulbs were selected from ten. Of the ten, four were defective. What is the probability that two of the three selected are defective?

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© 2002 Prentice-Hall, Inc. Chap 5-27 Hypergeometric Distribution Characteristics Mean Variance and Standard Deviation Finite Population Correction Factor

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© 2002 Prentice-Hall, Inc. Chap 5-28 Hypergeometric Distribution in PHStat PHStat | probability & prob. Distributions | Hypergeometric … Example in excel spreadsheet

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© 2002 Prentice-Hall, Inc. Chap 5-29 Chapter Summary Addressed the probability of a discrete random variable Defined covariance and discussed its application in finance Discussed binomial distribution Addressed Poisson distribution Discussed hypergeometric distribution

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