Business Statistics A First Course 3 rd Edition

Chapter Topics n The Probability of a Discrete Random Variable Covariance and its Applications in Finance n Binomial Distribution n The Normal Distribution n The Standardized Normal Distribution n Evaluating the Normality Assumption n © 2003 Prentice-Hall, Inc. 2

Random Variable n n Outcomes of an experiment expressed numerically e. g. Toss a

Discrete Random Variable n Obtained by Counting (0, 1, 2, 3, etc. ) n Usually a finite number of different values n e. g. Toss a coin 5 times; count the number of tails (0, 1, 2, 3, 4, or 5 times) © 2003 Prentice-Hall, Inc. 4

Discrete Probability Distribution Example Event: Toss 2 Coins. Count # Tails. Probability Distribution Values

Discrete Probability Distribution n List of All Possible [Xj , P(Xj) ] Pairs n Xj = Value of random variable n P(Xj) = Probability associated with value n Mutually Exclusive (Nothing in Common) n Collective Exhaustive (Nothing Left Out) © 2003 Prentice-Hall, Inc. 6

Summary Measures n Expected value (The Mean) n Weighted average of the probability distribution

Summary Measures n (continued) Variance n Weighted average squared deviation about the mean n

Covariance and its Application © 2003 Prentice-Hall, Inc. 9

Computing the Mean for Investment Returns Return per $1, 000 for two types of investments P(Xi) P(Yi) Investment Economic condition Dow Jones fund X Growth Stock Y . 2 Recession -$100 -$200 . 5 Stable Economy + 100 + 50 . 3 Expanding Economy + 250 + 350 © 2003 Prentice-Hall, Inc. 10

Computing the Variance for Investment Returns P(Xi) P(Yi) Investment Economic condition Dow Jones fund X Growth Stock Y . 2 Recession -$100 -$200 . 5 Stable Economy + 100 + 50 . 3 Expanding Economy + 250 + 350 © 2003 Prentice-Hall, Inc. 11

Computing the Covariance for Investment Returns P(Xi. Yi) Investment Economic condition Dow Jones fund X Growth Stock Y . 2 Recession -$100 -$200 . 5 Stable Economy + 100 + 50 . 3 Expanding Economy + 250 + 350 The Covariance of 23, 000 indicates that the two investments are positively related and will vary together in the same direction. © 2003 Prentice-Hall, Inc. 12

Important Discrete Probability Distributions Binomial © 2003 Prentice-Hall, Inc. 13

Binomial Probability Distribution n ‘n’ Identical Trials n n 2 Mutually Exclusive Outcomes on Each Trial n n e. g. 15 tosses of a coin; 10 light bulbs taken from a warehouse e. g. Head or tail in each toss of a coin; defective or not defective light bulb Trials are Independent n The outcome of one trial does not affect the outcome of the other © 2003 Prentice-Hall, Inc. 14

Binomial Probability Distribution (continued) n Constant Probability for Each Trial n n e. g. Probability of getting a tail is the same each time we toss the coin 2 Sampling Methods n n Infinite population without replacement Finite population with replacement © 2003 Prentice-Hall, Inc. 15

Binomial Probability Distribution Function Tails in 2 Tosses of Coin © 2003 Prentice-Hall, Inc.

Binomial Distribution Characteristics n Mean n E. g. Variance and Standard Deviation n .

Binomial Distribution in PHStat n n PHStat | Probability & Prob. Distributions | Binomial

Continuous Probability Distributions n Continuous Random Variable n n n Continuous Probability Distribution n n Values from interval of numbers Absence of gaps Distribution of continuous random variable Most Important Continuous Probability Distribution n The normal distribution © 2003 Prentice-Hall, Inc. 19

The Normal Distribution n n “Bell Shaped” Symmetrical Mean, Median and Mode are Equal Interquartile Range Equals 1. 33 Random Variable has Infinite Range © 2003 Prentice-Hall, Inc. f(X) X Mean Median Mode 20

The Mathematical Model © 2003 Prentice-Hall, Inc. 21

Many Normal Distributions There an Infinite Number of Normal Distributions Varying the Parameters and

Finding Probabilities Probability is the area under the curve! f(X) c © 2003 Prentice-Hall,

Which Table to Use? Infinitely Many Normal Distributions Mean Infinitely Many Tables to Look

Solution: The Cumulative Standardized Normal Distribution Table (Portion) Z . 00 . 01 . 02. 5478 0. 0. 5000. 5040. 5080 0. 1. 5398. 5438. 5478 0. 2. 5793. 5832. 5871 0. 3. 6179. 6217. 6255 © 2003 Prentice-Hall, Inc. Probabilities Z = 0. 12 Only One Table is Needed 25

Standardizing Example Normal Distribution © 2003 Prentice-Hall, Inc. Standardized Normal Distribution 26

Example: Normal Distribution © 2003 Prentice-Hall, Inc. Standardized Normal Distribution 27

Example: (continued) Cumulative Standardized Normal Distribution Table (Portion) Z . 00 . 01 . 02 . 5832 0. 0. 5000. 5040. 5080 0. 1. 5398. 5438. 5478 0. 2. 5793. 5832. 5871 0. 3. 6179. 6217. 6255 © 2003 Prentice-Hall, Inc. Z = 0. 21 28

Example: (continued) Cumulative Standardized Normal Distribution Table (Portion) Z . 00 . 01 . 02 . 4168 -0. 3. 3821. 3783. 3745 -0. 2. 4207. 4168. 4129 -0. 1. 4602. 4562. 4522 0. 0. 5000. 4960. 4920 © 2003 Prentice-Hall, Inc. Z = -0. 21 29

Normal Distribution in PHStat n n PHStat | Probability & Prob. Distributions | Normal

Example: Normal Distribution © 2003 Prentice-Hall, Inc. Standardized Normal Distribution 31

Example: (continued) Cumulative Standardized Normal Distribution Table (Portion) Z . 00 . 01 . 02 . 6179 0. 0. 5000. 5040. 5080 0. 1. 5398. 5438. 5478 0. 2. 5793. 5832. 5871 0. 3. 6179. 6217. 6255 © 2003 Prentice-Hall, Inc. Z = 0. 30 32

Finding Z Values for Known Probabilities What is Z Given Probability = 0. 6217 ? Cumulative Standardized Normal Distribution Table (Portion) Z . 6217 . 00 . 01 0. 2 0. 0. 5000. 5040. 5080 0. 1. 5398. 5438. 5478 0. 2. 5793. 5832. 5871 0. 3. 6179. 6217. 6255 © 2003 Prentice-Hall, Inc. 33

Recovering X Values for Known Probabilities Normal Distribution © 2003 Prentice-Hall, Inc. Standardized Normal

Assessing Normality n n Not All Continuous Random Variables are Normally Distributed It is Important to Evaluate how Well the Data Set Seems to be Adequately Approximated by a Normal Distribution © 2003 Prentice-Hall, Inc. 35

Assessing Normality n Construct Charts n n n (continued) For small- or moderate-sized data sets, do stemand-leaf display and box-and-whisker plot look symmetric? For large data sets, does the histogram or polygon appear bell-shaped? Compute Descriptive Summary Measures n n n Do the mean, median and mode have similar values? Is the interquartile range approximately 1. 33 ? Is the range approximately 6 ? © 2003 Prentice-Hall, Inc. 36

Assessing Normality n Observe the Distribution of the Data Set n n (continued) Do approximately between mean 2/3 of the observations lie 1 standard deviation? 4/5 of the observations lie 1. 28 standard deviations? 19/20 of the observations lie 2 standard deviations? Evaluate Normal Probability Plot n Do the points lie on or close to a straight line with positive slope? © 2003 Prentice-Hall, Inc. 37

Assessing Normality n (continued) Normal Probability Plot n n Arrange Data into Ordered Array Find Corresponding Standardized Normal Quantile Values Plot the Pairs of Points with Observed Data Values on the Vertical Axis and the Standardized Normal Quantile Values on the Horizontal Axis Evaluate the Plot for Evidence of Linearity © 2003 Prentice-Hall, Inc. 38

Assessing Normality (continued) Normal Probability Plot for Normal Distribution 90 X 60 Z 30

Normal Probability Plot Left-Skewed Right-Skewed 90 90 X 60 Z 30 -2 -1 0

Chapter Summary n n Addressed the Probability of a Discrete Random Variable Defined Covariance and Discussed its Application in Finance n Discussed Binomial Distribution n Discussed the Normal Distribution n Described the Standard Normal Distribution n Evaluated the Normality Assumption © 2003 Prentice-Hall, Inc. 41