Statistics for Managers Using Microsoft Excel/SPSS...Feb 04, 2016 · © 1999 Prentice-Hall, Inc....

© 1999 Prentice-Hall, Inc. Chap. 4 - 1

Statistics for Managers

Using Microsoft Excel/SPSS

Chapter 4

Basic Probability And Discrete Probability Distributions


Chapter Topics

• Basic Probability Concepts:

Sample Spaces and Events, Simple

Probability, and Joint Probability,

• Conditional Probability

• Bayes’ Theorem

• The Probability of a Discrete Random Variable

• Binomial, Poisson, and Hypergeometric

Distributions

• Covariance and its Applications in Finance


Sample Spaces

Collection of all Possible Outcomes

e.g. All 6 faces of a die:

e.g. All 52 cards of a bridge deck:


Events

• Simple Event: Outcome from a Sample Space

with 1 Characteristic

e.g. A Red Card from a deck of cards.

• Joint or Compound Event: Involves 2 Outcomes

Simultaneously

e.g. An Ace which is also a Red Card from a

deck of cards.

An Ace given that it is a Red Card.


Visualizing Events

•Contingency Tables

•Tree Diagrams

Ace Not Ace Total

Red 2 24 26

Black 2 24 26

Total 4 48 52


Simple Events

The Event of a Happy Face

There are 5 happy faces in this collection of 18 objects


Joint Events

The Event of a Happy Face AND Light Colored

3 Happy Faces which are light in color


12 Items, 5 happy faces and 7 other light objects

Compound Events

The Event of Happy Face OR Light Colored


Special Events

Null Event

Club & Diamond on

1 Card Draw

Complement of Event

For Event A,

All Events Not In A: A’

Null Event


3 Items, 3 Happy Faces Given they are Light Colored

Dependent or

Independent Events The Event of a Happy Face GIVEN it is Light Colored

E = Happy Face Light Color


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


Tree Diagram

Event Possibilities

Red

Cards

Black

Cards

Ace

Not an Ace

Ace

Not an Ace

Full Deck

of Cards


Probability

•Probability is the numerical

measure of the likelihood

that the event will occur.

•Value is between 0 and 1.

•Sum of the probabilities of

all mutually exclusive

events is 1.

Certain

Impossible

.5

1

0


Computing Probability

•The Probability of an Event, E:

•Each of the Outcome in the Sample Space

equally likely to occur.

SpaceSampleinOutcomesTotal

OutcomesEventofNumber)E(P

T

X

e.g. P( ) = 2/36

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


Computing

Joint Probability

The Probability of a Joint Event, A and B:

e.g. P(Red Card and Ace) =

CardsofNumberTotal

AcesdRe

52

2

26

1

P(A and B) =

Number of Event Outcomes from both A and B

Total Outcomes in Sample Space


P(A2 and B2)

P(A1 and B2)

P(A2 and B1)

P(A1 and B1)

Event

Event Total

Total 1

Joint Probability Using

Contingency Table

Joint Probability Marginal (Simple) Probability

P(A1) A1

A2

B1 B2

P(A2)

P(B1) P(B2)


Computing

Compound Probability

The Probability of a Compound Event, A or B:

SpaceSampleinOutcomesTotal

BorAeitherfromOutcomesEventofNumber)BorA(P

e.g.

P(Red Card or Ace)

CardsNumberofTotal

AcesdReCardsdReAces

52

2264

13

7

52

28


P(A1 and B1)

P(B2) P(B1)

P(A2 and B2) P(A2 and B1)

Event

Event Total

Total 1

Compound Probability

Addition Rule

P(A1 and B2) P(A1) A1

A2

B1 B2

P(A2)

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

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


Computing

Conditional Probability

The Probability of the Event:

Event A given that Event B has occurred

P(A B) =

e.g.

P(Red Card given that it is an Ace) =

)B(P

)BandA(P

2

1

4

2

Aces

AcesdRe


Black Color

Type Red Total

Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

Conditional Probability

Using Contingency Table

Conditional Event: Draw 1 Card. Note Kind & Color

26

2

5226

522

/

/

P(Red)

Red)AND P(Ace = Red) |P(Ace

Revised

Sample

Space


Conditional Probability and

Statistical Independence

)B(P

)BandA(PConditional Probability: P(AB) =

P(A and B) = P(A B) • P(B)

Events are

Independent:

P(A B) = P(A)

Or, P(A and B) = P(A) • P(B)

Events A and B are Independent when the probability

of one event, A is not affected by another event, B.

Multiplication Rule:


Bayes’ Theorem

)B(P)BA(P)B(P)BA(P

)B(P)BA(P

kk

ii

11

)A(P

)AandB(P i

P(Bi A) =

Adding up

the parts

of A in all

the B’s Same

Event


What are the chances of repaying a loan,

given a college education?

Bayes’ Theorem: Contingency Table

Loan Status

Education Repay Default Prob.

College .2 .05 .25

No College

Prob. 1

P(Repay College) = 08.

)DefaultandCollege(P)payReandCollege(P

)payReandCollege(P

? ? ?

? ?


Discrete Random Variable

• Random Variable: represents outcomes of an

experiment.

e.g. Throw a die twice:

Count the number of times 4 comes up (0, 1, or 2 times)

• Discrete Random Variable:

• Obtained by Counting (0, 1, 2, 3, etc.)

• Usually finite by number of values

e.g. Toss a coin 5 times. Count the number of tails.

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


Discrete Probability

Distribution Example

Probability Distribution

Values Probability

0 1/4 = .25

1 2/4 = .50

2 1/4 = .25

Event: Toss 2 Coins. Count # Tails.

T

T

T T


Discrete

Probability Distribution

• List of All Possible [ Xi, P(Xi) ] Pairs

Xi = Value of Random Variable (Outcome)

P(Xi) = Probability Associated with Value

• Mutually Exclusive (No Overlap)

• Collectively Exhaustive (Nothing Left

Out) 0 P(Xi) 1

S P(Xi) = 1


Discrete Random Variable

Summary Measures

Expected Value

The Mean of the Probability Distribution Weighted Average

m = E(X) = Xi P(Xi)

e.g. Toss 2 coins, Count tails, Compute Expected Value:

m = 0 .25 + 1 .50 + 2 .25 = 1

Variance

Weighted Average Squared Deviation about Mean

s2 = E [ (Xi - m )2]=S (Xi - m )2P(Xi)

e.g. Toss 2 coins, Count tails, Compute Variance:

s2 = (0 - 1)2(.25) + (1 - 1)2(.50) + (2 - 1)2(.25) = .50

Number of Tails


Important Discrete Probability

Distribution Models

Discrete Probability

Distributions

Binomial Hypergeometric Poisson


Binomial Probability Distributions

• ‘n’ Identical Trials, e.g. 15 tosses of a coin,

10 light bulbs taken from a warehouse

• 2 Mutually Exclusive Outcomes,

e.g. heads or tails in each toss of a coin,

defective or not defective light bulbs

• Constant Probability for each Trial,

e.g. probability of getting a tail is the same each time we toss the coin and each light bulb has the same probability of being defective


Binomial Probability

Distributions

• 2 Sampling Methods:

Infinite Population Without Replacement

Finite Population With Replacement

• Trials are Independent:

The Outcome of One Trial Does Not Affect the

Outcome of Another


Binomial Probability

Distribution Function

P(X) = probability that X successes given a knowledge of n

and p

X = number of ‘successes’ in

sample, (X = 0, 1, 2, ..., n)

p = probability of ‘success’

n = sample size

P(X) n

X ! n X p p

X n X !

( ) ! ( )

1

Tails in 2 Toss of Coin

X P(X)

0 1/4 = .25

1 2/4 = .50

2 1/4 = .25


Binomial Distribution

Characteristics

n = 5 p = 0.1

n = 5 p = 0.5

Mean

Standard Deviation

m

s

E X np

np p

( )

( )

0

.2

.4

.6

0 1 2 3 4 5

X

P(X)

.2

.4

.6

0 1 2 3 4 5

X

P(X)

e.g. m = 5 (.1) = .5

e.g. s = 5(.5)(1 - .5)

= 1.118

0


Poisson Distribution

Poisson Process:

• Discrete Events in an ‘Interval’

The Probability of One Success in

Interval is Stable

The Probability of More than One

Success in this Interval is 0

• Probability of Success is

Independent from Interval to

Interval

e.g. # Customers Arriving in 15 min.

# Defects Per Case of Light

Bulbs.

P X x

x

x

( |

!

l

l l e

-


Poisson Probability

Distribution Function

P(X ) = probability of X successes given l

l = expected (mean) number of ‘successes’

e = 2.71828 (base of natural logs)

X = number of ‘successes’ per unit

P X X

X

( ) !

l l

e

e.g. Find the probability of 4

customers arriving in 3 minutes

when the mean is 3.6.

P(X) = e -3.6

3.6

4!

4

= .1912


Poisson Distribution

Characteristics

l = 0.5

l = 6

Mean

Standard Deviation

m l

s l

i i

N

i

E X

X P X

( )

( ) 1

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)


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


Hypergeometric Distribution

P(X) = probability that X successes given n, N, and A

n = sample size

N = population size

A = number of “successes”

in population

X = number of “successes”

in sample (X = 0, 1, 2, ..., n)

P X ) ( )( ) A

X (

( )

N - A

n - X

N

n

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?

P(2) = ( )( ) 4

2

6

1 10

3 ( )

= .30

(


Hypergeometric Characteristics

Mean

Standard Deviation

m

s

E X n

nA N A

( )

N n

N 1

Finite

Population

Correction

A

N

N 2

)(


Covariance

X = discrete random variable X

Xi = ith outcome of X

P(XiYi) = probability of occurrence of the

ith outcome of Y

Y = discrete random variable Y

Yi = ith outcome of Y

i = 1, 2, …, N

)YX(P)Y(EY)X(EX iii

N

iiXY

1

s


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

E(X) = mX = (-100)(.2) + (100)(.5) + (250)(.3) = $105

E(Y) = mY = (-200)(.2) + (50)(.5) + (350)(.3) = $90


Computing the Variance for Investment Returns


.2 Recession -$100 -$200



Investment

Var(X) = = (.2)(-100 -105)2 + (.5)(100 - 105)2 + (.3)(250 - 105)2

= 14,725, sX = 121.35

Var(Y) = = (.2)(-200 - 90)2 + (.5)(50 - 90)2 + (.3)(350 - 90)2

= 37,900, sY = 194.68

2Xs

2Ys


Computing the Covariance for Investment Returns


.2 Recession -$100 -$200



Investment

sXY = (.2)(-100 - 105)(-200 - 90) + (.5)(100 - 105)(50 - 90)

+ (.3)(250 -105)(350 - 90) = 23,300

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

strongly related and will vary together in the same direction.


Chapter Summary

•Discussed Basic Probability Concepts:

Sample Spaces and Events, Simple Probability,

and Joint Probability

•Defined Conditional Probability

•Discussed Bayes’ Theorem

•Addressed the Probability of a Discrete Random Variable

•Discussed Binomial, Poisson, and Hypergeometric Distributions

• Addressed Covariance and its Applications in Finance

Statistics for Managers Using Microsoft Excel/SPSS...Feb 04, 2016 · © 1999 Prentice-Hall, Inc....

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Transcript of Statistics for Managers Using Microsoft Excel/SPSS...Feb 04, 2016 · © 1999 Prentice-Hall, Inc....