Statistics for Managers Using Microsoft Excel,...

© 2002 Prentice-Hall, Inc. Chap 4-1

Statistics for Managers Using Microsoft Excel

(3rd Edition)

Chapter 4

Basic Probability and Discrete Probability Distributions

© 2002 Prentice-Hall, Inc.

Chap 4-2

Chapter Topics

Basic probability concepts

Sample spaces and events, simple probability, joint

probability

Conditional probability

Statistical independence, marginal probability

Bayes’s Theorem


Chap 4-3

Chapter Topics

The probability of a discrete random variable

Covariance and its applications in finance

Binomial distribution

Poisson distribution

Hypergeometric distribution

(continued)


Chap 4-4

Sample Spaces

Collection of all possible outcomes

e.g.: All six faces of a die:

e.g.: All 52 cards in a deck:


Chap 4-5

Events

Simple event

Outcome from a sample space with one

characteristic

e.g.: A red card from a deck of cards

Joint event

Involves two outcomes simultaneously

e.g.: An ace that is also red from a deck of

cards


Chap 4-6

Visualizing Events

Contingency Tables

Tree Diagrams

Red 2 24 26

Black 2 24 26

Total 4 48 52

Ace Not Ace Total

Full Deck

of Cards

Red

Cards

Black

Cards

Not an Ace

Ace

Ace

Not an Ace


Chap 4-7

Simple Events

The Event of a Triangle

There are 5 triangles in this collection of 18 objects


Chap 4-8

The event of a triangle AND blue in color

Joint Events

Two triangles that are blue


Chap 4-9

Special Events

Impossible event

e.g.: Club & diamond on one card draw

Complement of event

For event A, all events not in A

Denoted as A’

e.g.: A: queen of diamonds A’: all cards in a deck that are not queen of diamonds

Null Event


Chap 4-10

Special Events

Mutually exclusive events Two events cannot occur together

e.g.: -- A: queen of diamonds; B: queen of clubs

Events A and B are mutually exclusive Collectively exhaustive events

One of the events must occur

The set of events covers the whole sample space

e.g.: -- A: all the aces; B: all the black cards; C: all the diamonds; D: all the hearts

Events A, B, C and D are collectively exhaustive

Events B, C and D are also collectively exhaustive

(continued)


Chap 4-11

Contingency Table

A Deck of 52 Cards

Ace Not an

Ace Total

Red

Black

Total

2 24

2 24

26

26

4 48 52

Sample Space

Red Ace


Chap 4-12

Full Deck

of Cards

Tree Diagram

Event Possibilities

Red

Cards

Black

Cards

Ace

Not an Ace

Ace

Not an Ace


Chap 4-13

Probability

Probability is the numerical

measure of the likelihood

that an event will occur

Value is between 0 and 1

Sum of the probabilities of

all mutually exclusive and

collective exhaustive events

is 1

Certain

Impossible

.5

1

0


Chap 4-14

(There are 2 ways to get one 6 and the other 4)

e.g. P( ) = 2/36

Computing Probabilities

The probability of an event E:

Each of the outcomes in the sample space is

equally likely to occur

number of event outcomes( )

total number of possible outcomes in the sample space

P E

X

T


Chap 4-15

Computing Joint Probability

The probability of a joint event, A and B:

( and ) = ( )

number of outcomes from both A and B

total number of possible outcomes in sample space

P A B P A B

E.g. (Red Card and Ace)

2 Red Aces 1

52 Total Number of Cards 26

P


Chap 4-16

P(A1 and B2) P(A1)

Total Event

Joint Probability Using Contingency Table

P(A2 and B1)

P(A1 and B1)

Event

Total 1

Joint Probability Marginal (Simple) Probability

A1

A2

B1 B2

P(B1) P(B2)

P(A2 and B2) P(A2)


Chap 4-17

Computing Compound Probability

Probability of a compound event, A or B:

( or ) ( )

number of outcomes from either A or B or both

total number of outcomes in sample space

P A B P A B

E.g. (Red Card or Ace)

4 Aces + 26 Red Cards - 2 Red Aces

52 total number of cards

28 7

52 13

P


Chap 4-18

P(A1)

P(B2)

P(A1 and B1)

Compound Probability (Addition Rule)

P(A1 or B1 ) = P(A1) + P(B1) - P(A1 and B1)

P(A1 and B2)

Total Event

P(A2 and B1)

Event

Total 1

A1

A2

B1 B2

P(B1)

P(A2 and B2) P(A2)

For Mutually Exclusive Events: P(A or B) = P(A) + P(B)


Chap 4-19

Computing Conditional Probability

The probability of event A given that event B has occurred:

( and )( | )

( )

P A BP A B

P B

E.g.

(Red Card given that it is an Ace)

2 Red Aces 1

4 Aces 2

P


Chap 4-20

Conditional Probability Using Contingency Table

Black

Color Type Red Total

Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

Revised Sample Space

(Ace and Red) 2 / 52 2(Ace | Red)

(Red) 26 / 52 26

PP

P


Chap 4-21

Conditional Probability and Statistical Independence

Conditional probability:

Multiplication rule:

( and )( | )

( )

P A BP A B

P B

( and ) ( | ) ( )

( | ) ( )

P A B P A B P B

P B A P A


Chap 4-22

Conditional Probability and Statistical Independence

Events A and B are independent if

Events A and B are independent when the probability of one event, A, is not affected by another event, B

(continued)

( | ) ( )

or ( | ) ( )

or ( and ) ( ) ( )

P A B P A

P B A P B

P A B P A P B


Chap 4-23

Bayes’s Theorem

1 1

||

| |

and

i i

i

k k

i

P A B P BP B A

P A B P B P A B P B

P B A

P A

Adding up

the parts

of A in all

the B’s Same

Event


Chap 4-24

Bayes’s Theorem Using Contingency Table

Fifty percent of borrowers repaid their loans. Out of those who repaid, 40% had a college degree. Ten percent of those who defaulted had a college degree. What is the probability that a randomly selected borrower who has a college degree will repay the loan?

.50 | .4 | .10P R P C R P C R

| ?P R C


Chap 4-25

Bayes’s Theorem Using Contingency Table

||

| |

.4 .5 .2 .8

.4 .5 .1 .5 .25

P C R P RP R C

P C R P R P C R P R

(continued)

Repay Repay

College

College

1.0 .5 .5

.2

.3

.05

.45

.25

.75

Total

Total


Chap 4-26

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)


Chap 4-27

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)


Chap 4-28

Probability Distribution

Values Probability

0 1/4 = .25

1 2/4 = .50

2 1/4 = .25

Discrete Probability

Distribution Example

T

T

T T

Event: Toss two coins Count the number of tails


Chap 4-29

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)

Collectively exhaustive (nothing left out)

0 1 1j jP X P X


Chap 4-30

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 j

j

E X X P X

0 2.5 1 .5 2 .25 1

j j

j

X P X


Chap 4-31

Summary Measures

Variance

Weight average squared deviation about the mean

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

(continued)

222

j jE X X P X

22

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

j jX P X


Chap 4-32

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 i

i

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


Chap 4-33

Computing the Mean for

Investment Returns

Return per $1,000 for two types of investments

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

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


Chap 4-34

Computing the Variance for

Investment Returns


.2 Recession -$100 -$200



Investment

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

14,725 121.35

X

X

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

37,900 194.68

Y

Y


Chap 4-35

Computing the Covariance for

Investment Returns


.2 Recession -$100 -$200



Investment

The Covariance of 23,000 indicates that the two investments are

positively related and will vary together in the same direction.

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

250 105 350 90 .3 23,300

XY


Chap 4-36

Important Discrete Probability Distributions

Discrete Probability

Distributions

Binomial Hypergeometric Poisson


Chap 4-37

Binomial Probability Distribution

‘n’ identical trials e.g.: 15 tosses of a coin; ten light bulbs taken

from a warehouse

Two mutually exclusive outcomes on each trials 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


Chap 4-38

Binomial Probability Distribution

Constant probability for each trial

e.g.: Probability of getting a tail is the same each time we toss the coin

Two sampling methods

Infinite population without replacement

Finite population with replacement

(continued)


Chap 4-39

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

!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


Chap 4-40

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


Chap 4-41

Binomial Distribution in PHStat

PHStat | probability & prob. Distributions | binomial

Example in excel spreadsheet


Chap 4-42

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.: number of customers arriving in 15 minutes

e.g.: number of defects per case of light bulbs

P X x

x

x

( |

!

e -


Chap 4-43

Poisson Probability Distribution Function

e.g.: Find the probability of 4

customers arriving in 3 minutes

when the mean is 3.6.

3.6 43.6

.19124!

eP X

!

: probability of "successes" given

: number of "successes" per unit

: expected (average) number of "successes"

: 2.71828 (base of natural logs)

XeP X

X

P X X

X

e


Chap 4-44

Poisson Distribution in PHStat

PHStat | probability & prob. Distributions | Poisson



Chap 4-45

Poisson Distribution Characteristics

Mean

Standard Deviation and Variance

1

N

i i

i

E X

X P X

= 0.5

= 6

0

.2

.4

.6

0 1 2 3 4 5

X

P(X)

0

.2

.4

.6

0 2 4 6 8 10

X

P(X)

2


Chap 4-46

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


Chap 4-47


Function

E.g. 3 Light bulbs were

selected from 10. Of the 10

there were 4 defective. What

is the probability that 2 of the

3 selected are defective?

4 6

2 12 .30

10

3

P

: probability that successes given , , and

: sample size

: population size

: number of "successes" in population

: number of "successes" in sample

0,1,2,

A N A

X n XP X

N

n

P X X n N A

n

N

A

X

X

,n


Chap 4-48


Characteristics

Mean

Variance and Standard Deviation

A

E X nN

2

2

2

1

1

nA N A N n

N N

nA N A N n

N N

Finite Population Correction Factor


Chap 4-49

Hypergeometric Distribution in PHStat

PHStat | probability & prob. Distributions | Hypergeometric …



Chap 4-50

Chapter Summary

Discussed basic probability concepts

Sample spaces and events, simple

probability, and joint probability

Defined conditional probability

Statistical independence, marginal probability

Discussed Bayes’s theorem


Chap 4-51

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

(continued)

Date post:	09-May-2019
Category:	Documents
Upload:	vudien
View:	224 times
Download:	0 times

Statistics for Managers Using Microsoft Excel,...

Documents