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

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Presentation on theme: "Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.. Chap 5-1 Chapter 5 Some Important Discrete Probability Distributions Basic Business Statistics."— Presentation transcript:

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

2 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, Poisson distributions and hypergeometric  How to use the binomial, hypergeometric, and Poisson distributions to solve business problems

3 Basic Business Statistics, 10e © 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

4 Basic Business Statistics, 10e © 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)

5 Basic Business Statistics, 10e © 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 X X Value Probability 0 1/4 = /4 = /4 = Probability

6 Basic Business Statistics, 10e © 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)

7 Basic Business Statistics, 10e © 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)

8 Basic Business Statistics, 10e © 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

9 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-9 二元隨機變數之機率分配

10 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-10 事件機率

11 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-11 表 5.6 一般化的聯合機率分配表

12 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-12 Expected Value

13 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-13 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

14 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-14 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

15 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-15 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

16 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-16 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

17 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-17 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

18 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-18 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:

19 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-19 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

20 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-20 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

21 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-21 Probability Distributions Continuous Probability Distributions Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions Normal Uniform Exponential Ch. 5Ch. 6

22 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-22 The Binomial Distribution Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions

23 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  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

24 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-24 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

25 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-25 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

26 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-26 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)

27 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-27 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 = ( ) = 0.5 X = 0, 1, 2, 3, 4 Binomial Distribution Formula

28 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-28 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

29 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-29 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

30 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-30 Binomial Distribution Characteristics  Mean  Variance and Standard Deviation Wheren = sample size p = probability of success (1 – p) = probability of failure

31 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-31

32 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-32

33 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-33

34 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-34

35 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-35 The Bernoulli Distribution  Suppose that a random experiment can give rise to just two possible mutually exclusive and collectively exhaustive outcomes, which for convenience we will label “ success ” and “ failure. ” Let p denote the probability of success, so that the probability of failure is (1-p). Now define the random variable X so that X takes the value 1 if the outcome of the experiment is success and 0 otherwise.  The probability function of this random variable is then P x (0) = (1-p), P x (1) = p, This distribution is known as the Bernoulli distribution. Its mean and variance are E(X) = 0*(1-p)+1*(p) = p V(X) = (0-p) 2 (1-p)+(1-p) 2 p = p(1-p)

36 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-36 Binomial Distribution  Experiment involves n identical trials  Each trial has exactly two possible outcomes: success and failure  Each trial is independent of the previous trials  p is the probability of a success on any one trial  q = (1-p) is the probability of a failure on any one trial  p and q are constant throughout the experiment  X is the number of successes in the n trials  Applications  Sampling with replacement  Sampling without replacement -- n < 5% N

37 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-37 n = 5 p = 0.1 n = 5 p = 0.5 Mean X P(X) X P(X) 0 Binomial Characteristics Examples

38 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-38 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) =

39 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-39 The Hypergeometric Distribution Binomial Poisson Probability Distributions Discrete Probability Distributions Hypergeometric

40 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-40 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

41 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-41 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

42 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-42 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

43 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-43 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%.

44 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-44 The Poisson Distribution Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions

45 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-45 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)

46 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-46 Poisson Distribution Formula where: X = number of events in an area of opportunity = expected number of events e = base of the natural logarithm system ( )

47 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-47 Poisson Distribution Characteristics  Mean  Variance and Standard Deviation where = expected number of events

48 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-48 Using Poisson Tables X Example: Find P(X = 2) if = 0.50

49 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-49 Graph of Poisson Probabilities X = P(X = 2) = Graphically: = 0.50

50 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-50 Poisson Distribution Shape  The shape of the Poisson Distribution depends on the parameter : = 0.50 = 3.00

51 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-51

52 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-52

53 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-53 表 6.21 二項分配與泊松分 配

54 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-54 圖 6.15 二項分配 圖 6.16 泊松 分配

55 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-55 圖 6.17 二項分配與泊松分配

56 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-56 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  Discussed the Poisson distribution


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