© 2003 Prentice-Hall, Inc.Chap 5-1 Business Statistics: A First Course (3 rd Edition) Chapter 5...

© 2003 Prentice-Hall, Inc. Chap 5-1

Business Statistics: A First Course

(3rd Edition)

Chapter 5Probability Distributions

© 2003 Prentice-Hall, Inc. Chap 5-2

Chapter Topics

The Probability of a Discrete Random Variable

Covariance and its Applications in Finance

Binomial Distribution The Normal Distribution The Standardized Normal Distribution Evaluating the Normality Assumption

© 2003 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)

© 2003 Prentice-Hall, Inc. Chap 5-4

Discrete Random Variable

Discrete Random Variable Obtained by Counting (0, 1, 2, 3, etc.) Usually a finite number of different values e.g. Toss a coin 5 times; count the number

of tails (0, 1, 2, 3, 4, or 5 times)

© 2003 Prentice-Hall, Inc. Chap 5-5

Probability Distribution Values Probability

0 1/4 = .25

1 2/4 = .50

2 1/4 = .25

Discrete Probability Distribution Example

Event: Toss 2 Coins. Count # Tails.

T

T

T T

© 2003 Prentice-Hall, Inc. Chap 5-6

Discrete Probability Distribution

List of All Possible [Xj , P(Xj) ] Pairs

Xj = Value of random variable

P(Xj) = Probability associated with value

Mutually Exclusive (Nothing in Common)

Collective Exhaustive (Nothing Left Out)

0 1 1j jP X P X

© 2003 Prentice-Hall, Inc. Chap 5-7

Summary Measures

Expected value (The Mean) Weighted average of the probability

distribution

e.g. Toss 2 coins, count the number of tails, compute expected value

j jj

E X X P X

0 .25 1 .5 2 .25 1

j jj

X P X

© 2003 Prentice-Hall, Inc. Chap 5-8

Summary Measures

Variance Weighted average squared deviation about the

mean

e.g. Toss 2 coins, count number of tails, compute variance

(continued)

222j jE X X P X

22

2 2 2 0 1 .25 1 1 .5 2 1 .25 .5

j jX P X

© 2003 Prentice-Hall, Inc. Chap 5-9

Covariance and its Application

1

th

th

th

: discrete random variable

: outcome of

: discrete random variable

: outcome of

: probability of occurrence of the

outcome of an

N

XY i i i ii

i

i

i i

X E X Y E Y P X Y

X

X i X

Y

Y i Y

P X Y i

X

thd the outcome of Yi

© 2003 Prentice-Hall, Inc. Chap 5-10

Computing the Mean for Investment Returns

Return per $1,000 for two types of investments

P(Xi) P(Yi) Economic condition Dow Jones fund X Growth Stock Y

.2 .2 Recession -$100 -$200

.5 .5 Stable Economy + 100 + 50

.3 .3 Expanding Economy + 250 + 350

Investment

100 .2 100 .5 250 .3 $105XE X

200 .2 50 .5 350 .3 $90YE Y

© 2003 Prentice-Hall, Inc. Chap 5-11

Computing the Variance for Investment Returns

2 2 22 .2 100 105 .5 100 105 .3 250 105

14,725 121.35X

X

2 2 22 .2 200 90 .5 50 90 .3 350 90

37,900 194.68Y

Y

P(Xi) P(Yi) Economic condition Dow Jones fund X Growth Stock Y

.2 .2 Recession -$100 -$200

.5 .5 Stable Economy + 100 + 50

.3 .3 Expanding Economy + 250 + 350

Investment

© 2003 Prentice-Hall, Inc. Chap 5-12

Computing the Covariance for Investment Returns

P(XiYi) Economic condition Dow Jones fund X Growth Stock Y

.2 Recession -$100 -$200

.5 Stable Economy + 100 + 50

.3 Expanding Economy + 250 + 350

Investment

100 105 200 90 .2 100 105 50 90 .5

250 105 350 90 .3 23,300

XY

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. Chap 5-13

Important Discrete Probability Distributions

Discrete Probability Distributions

Binomial

© 2003 Prentice-Hall, Inc. Chap 5-14

Binomial Probability Distribution

‘n’ Identical Trials e.g. 15 tosses of a coin; 10 light bulbs taken

from a warehouse 2 Mutually Exclusive Outcomes on Each

Trial e.g. Head or tail 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

© 2003 Prentice-Hall, Inc. Chap 5-15

Binomial Probability Distribution

Constant Probability for Each Trial e.g. Probability of getting a tail is the same

each time we toss the coin 2 Sampling Methods

Infinite population without replacement Finite population with replacement

(continued)

© 2003 Prentice-Hall, Inc. Chap 5-16

Binomial Probability Distribution Function

!1

! !

: probability of successes given and

: number of "successes" in sample 0,1, ,

: the probability of each "success"

: sample size

n XXnP X p p

X n X

P X X n p

X X n

p

n

Tails in 2 Tosses of Coin

X P(X) 0 1/4 = .25

1 2/4 = .50

2 1/4 = .25

© 2003 Prentice-Hall, Inc. Chap 5-17

Binomial Distribution Characteristics

Mean E.g.

Variance and Standard Deviation

e.g.

E X np 5 .1 .5np

n = 5 p = 0.1

0.2.4.6

0 1 2 3 4 5

X

P(X)

1 5 .1 1 .1 .6708np p

2 1

1

np p

np p

© 2003 Prentice-Hall, Inc. Chap 5-18

Binomial Distribution in PHStat

PHStat | Probability & Prob. Distributions | Binomial

Example in Excel Spreadsheet

Microsoft Excel Worksheet

© 2003 Prentice-Hall, Inc. Chap 5-19

Continuous Probability Distributions

Continuous Random Variable Values from interval of numbers Absence of gaps

Continuous Probability Distribution Distribution of continuous random variable

Most Important Continuous Probability Distribution The normal distribution

© 2003 Prentice-Hall, Inc. Chap 5-20

The Normal Distribution

“Bell Shaped” Symmetrical Mean, Median and

Mode are Equal Interquartile Range

Equals 1.33 Random Variable

has Infinite Range

Mean Median Mode

X

f(X)

© 2003 Prentice-Hall, Inc. Chap 5-21

The Mathematical Model

2(1/ 2) /1

2

: density of random variable

3.14159; 2.71828

: population mean

: population standard deviation

: value of random variable

Xf X e

f X X

e

X X

© 2003 Prentice-Hall, Inc. Chap 5-22

Many Normal Distributions

Varying the Parameters and , we obtain Different Normal Distributions

There are an Infinite Number of Normal Distributions

© 2003 Prentice-Hall, Inc. Chap 5-23

Finding Probabilities

Probability is the area under the curve!

c dX

f(X)

?P c X d

© 2003 Prentice-Hall, Inc. Chap 5-24

Which Table to Use?

Infinitely Many Normal Distributions Mean Infinitely Many Tables to Look

Up!

© 2003 Prentice-Hall, Inc. Chap 5-25

Solution: The Cumulative Standardized Normal

Distribution

Z .00 .01

0.0 .5000 .5040 .5080

.5398 .5438

0.2 .5793 .5832 .5871

0.3 .6179 .6217 .6255

.5478.02

0.1 .5478

Cumulative Standardized Normal Distribution Table (Portion)

Probabilities

Only One Table is Needed

0 1Z Z

Z = 0.12

0

© 2003 Prentice-Hall, Inc. Chap 5-26

Standardizing Example

6.2 50.12

10

XZ

Normal Distribution

Standardized Normal

Distribution10 1Z

5 6.2 X Z

0Z 0.12

© 2003 Prentice-Hall, Inc. Chap 5-27

Example:

Normal Distribution

Standardized Normal

Distribution10 1Z

5 7.1 X Z0Z

0.21

2.9 5 7.1 5.21 .21

10 10

X XZ Z

2.9 0.21

.0832

2.9 7.1 .1664P X

.0832

© 2003 Prentice-Hall, Inc. Chap 5-28

Z .00 .01

0.0 .5000 .5040 .5080

.5398 .5438

0.2 .5793 .5832 .5871

0.3 .6179 .6217 .6255

.5832.02

0.1 .5478

Cumulative Standardized Normal Distribution Table (Portion)

0 1Z Z

Z = 0.21

Example: 2.9 7.1 .1664P X

(continued)

0

© 2003 Prentice-Hall, Inc. Chap 5-29

Z .00 .01

-0.3 .3821 .3783 .3745

.4207 .4168

-0.1.4602 .4562 .4522

0.0 .5000 .4960 .4920

.4168.02

-0.2 .4129

Cumulative Standardized Normal Distribution Table (Portion)

0 1Z Z

Z = -0.21

Example: 2.9 7.1 .1664P X

(continued)

0

© 2003 Prentice-Hall, Inc. Chap 5-30

Normal Distribution in PHStat

PHStat | Probability & Prob. Distributions | Normal …

Example in Excel Spreadsheet

Microsoft Excel Worksheet

© 2003 Prentice-Hall, Inc. Chap 5-31

Example: 8 .3821P X

Normal Distribution

Standardized Normal

Distribution10 1Z

5 8 X Z0Z

0.30

8 5.30

10

XZ

.3821

© 2003 Prentice-Hall, Inc. Chap 5-32

Example: 8 .3821P X

(continued)

Z .00 .01

0.0 .5000 .5040 .5080

.5398 .5438

0.2 .5793 .5832 .5871

0.3 .6179 .6217 .6255

.6179.02

0.1 .5478

Cumulative Standardized Normal Distribution Table (Portion)

0 1Z Z

Z = 0.30

0

© 2003 Prentice-Hall, Inc. Chap 5-33

.6217

Finding Z Values for Known Probabilities

Z .00 0.2

0.0 .5000 .5040 .5080

0.1 .5398 .5438 .5478

0.2 .5793 .5832 .5871

.6179 .6255

.01

0.3

Cumulative Standardized Normal Distribution Table

(Portion)

What is Z Given Probability = 0.6217 ?

.6217

0 1Z Z

.31Z 0

© 2003 Prentice-Hall, Inc. Chap 5-34

Recovering X Values for Known Probabilities

5 .30 10 8X Z

Normal Distribution

Standardized Normal

Distribution10 1Z

5 ? X Z0Z 0.30

.3821.6179

© 2003 Prentice-Hall, Inc. Chap 5-35

Assessing Normality

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. Chap 5-36

Assessing Normality Construct Charts

For small- or moderate-sized data sets, do stem-and-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 Do the mean, median and mode have similar

values? Is the interquartile range approximately 1.33

? Is the range approximately 6 ?

(continued)

© 2003 Prentice-Hall, Inc. Chap 5-37

Assessing Normality

Observe the Distribution of the Data Set Do approximately 2/3 of the observations lie

between mean 1 standard deviation? Do approximately 4/5 of the observations lie

between mean 1.28 standard deviations? Do approximately 19/20 of the observations

lie between mean 2 standard deviations? Evaluate Normal Probability Plot

Do the points lie on or close to a straight line with positive slope?

(continued)

© 2003 Prentice-Hall, Inc. Chap 5-38

Assessing Normality

Normal Probability Plot 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

(continued)

© 2003 Prentice-Hall, Inc. Chap 5-39

Assessing Normality

Normal Probability Plot for Normal Distribution

Look for a Straight Line!

30

60

90

-2 -1 0 1 2

Z

X

(continued)

© 2003 Prentice-Hall, Inc. Chap 5-40

Normal Probability Plot

Left-Skewed Right-Skewed

Rectangular U-Shaped

30

60

90

-2 -1 0 1 2

Z

X

30

60

90

-2 -1 0 1 2

Z

X

30

60

90

-2 -1 0 1 2

Z

X

30

60

90

-2 -1 0 1 2

Z

X

© 2003 Prentice-Hall, Inc. Chap 5-41

Chapter Summary

Addressed the Probability of a Discrete Random Variable

Defined Covariance and Discussed its Application in Finance

Discussed Binomial Distribution

Discussed the Normal Distribution

Described the Standard Normal Distribution

Evaluated the Normality Assumption