Lec 14 Discrete Prob Dist SLIDE

8/14/2019 Lec 14 Discrete Prob Dist SLIDE

1/17

6 - 1

Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.


2/17

6 - 2

TerminologyRandom Variable

is a numerical value determined by

the outcome of an experiment.

is a numerical value determined by

the outcome of an experiment.


Probability Distribution

is the listing ofall possible outcomes

of an experimentand thecorresponding probability.

is the listing ofall possible outcomes

of an experimentand thecorresponding probability.

Example


3/17

6 - 3

Tree Diagrams

This is a useful device to show all the possible outcomes

of the experimentand their corresponding probabilities


ons er ng e ran om exper men o pp ng a co n r ce.


4/17

6 - 4

In a random experiment in which

a coin is tossed three times:

In a random experiment in which

a coin is tossed three times:

HeadsLetx be the number of


TailsLet T represent the outcome of

LetHrepresent the outcome of a Head

Determine the probability distributionDetermine the probability distribution


5/17

6 - 5

Tree DiagramsOrigin First

Flip

HH

SecondFlip

H

HT HHT

HTH

HHH

ThirdFlip


T

TH

T

TTT

H

H

T

T

T

THH

THTTTH


6/17

6 - 6Listing the possibilitiesListing the possibilities

Heads Heads Heads TailsHeads Heads

TailsHeads Heads Tails Heads Heads


TailsHeads Tails TailsHeadsTails

TailsTails Tails

the possible values ofx

(number of heads) are 0,1,2,3.

the possible values ofx

(number of heads) are 0,1,2,3.

Tails HeadsTails


7/17

6 - 7

DiscreteDiscrete

Types ofProbability Distributions

ContinuousContinuous


the random variablehas a

countable numberof possible outcomes

the random variablehas a

countable numberof possible outcomes

the random variablehas an

infinite numberof possible outcomes

the random variablehas an

infinite numberof possible outcomes

ExamplesExamples


8/17

6 - 8

DiscreteDiscrete ContinuousContinuous

ExamplesExamplesStudents in a classStudents in a class

Types ofProbability Distributions

Distance driven by an


Number ofchildren

in a family

Mortgage Loan

Number of Mortgages

approved in a month

executive to get to work

The length of time of a

particular phone call

The length of

time of an

afternoon nap!


9/17


10/17

6 - 10

Consider a

random

experiment in

which acoin is

tossed three times.

Consider a

random

experiment in

which acoin is

tossed three times.

Probability DistributionProbability Distribution

x # ofOutcomes

P(x)

0 1 1/8


Determine theprobability

distribution.

Determine theprobability

distribution.

1

2

3

3

3

1

8

3/8

3/8

1/8

8/8 = 1

What is theprobability of

tossing 2 heads in

3 flips?


11/17

6 - 11

reports thecentral location of the data

Mean of a Discrete

ProbabilityDistribution

Mean of a Discrete


is denoted by the Greek symbol ,mu


is the long- un average value ofthe random variable

also referred to as its expected value,E(X),

in a probability distribution


12/17

6 - 12

xP(x)

Mean of a Discrete


Mean of a Discrete


Fli a c in thr tim

)]([xxP=

)(

xxPFormulaFormula

x P(x)


Letx be thenumber of heads

Letx be thenumber of heads

3/8

3/8

6/8

12/8=1.5

1

2

3

3/8

3/8

1/8

8/8 = 1


13/17


14/17

6 - 14

Varianceof a Discrete


Varianceof a Discrete


Flip a coin three times.

Letx be the number of heads

Flip a coin three times.

Letx be the number of heads

)]()[(2

xPx 2 =

(x)Px FormulaFormula


(X- )2X-- 1.5

- 0.5

1.50.5

0.25

2.25

0.25

(X- )2 P(x).28125

.09375

0.75

.09375

.28125

xP(x)0

3/8

3/8

6/8

x0

1

2

3

P(x)1/8

3/8

3/8

1/8

8/8 = 1 =1.5

2.25


15/17

6 - 15

Dan Desch, owner

of College Painters,

studied his recordsfor the past

20 weeks

and re orted the

Dan Desch, owner

of College Painters,

studied his recordsfor the past

20 weeks

and re orted the

P(x)

5/20 = 0.25

6/20 = 0.30

# of Painted

Houses

10

11

Weeks

5

6

x


followingnumber of houses

painted per week:

followingnumber of houses

painted per week:

7/20 = 0.35

2/20 = 0.10

20/20 = 1.0

Computing theComputing the

Determine theProbability distribution and itsmean and variance.Determine theProbability distribution and itsmean and variance.

12

13

7

2

20


16/17

6 - 16

)]([ xxP= )(xxPxP(x)

2.5

Computing theComputing the

FormulaFormula

P(x)# of PaintedHouses

5/20 = 0.2510


3.3

1.3

4.2

11.3

Computing theComputing the2

6/20 = 0.30

7/20 = 0.35

2/20 = 0.10

20/20 = 1.0

11

12

13


17/17

6 - 17

)]()[(2

xPx 2 =

(x - )2

1.69

(x - )2 P(x)

.4225

(x)P

Computing theComputing the 2

P(x)

0.25

x

10

xP(x)

2.5

)2

(x FormulaFormula

(10-11.3)2(10-11.3)2


0.09

2.89

0.49

.0270

0.910.91

.1715

.2890

0.30

0.35

0.10

1.0

11

12

13

3.3

1.3

4.2

11.3

Documents

Lec 14 Discrete Prob Dist SLIDE