OPRE 6301-SYSM 6303 Chapter 07 Slides_students

8/17/2019 OPRE 6301-SYSM 6303 Chapter 07 Slides_students

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OPRE 6301/SYSM 6303Quantitative Introduction to Risk and

Uncertainty in Business

7-1

Chapter SevenRandom Variables

andDiscrete Probability Distributions

7-2


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Random Variable

A random variable is

a function or rule

that assigns a number

to each outcome of an experiment

7-3

Random Variable

Discrete Random Variable

Variable who can take on

a countable number of values

Continuous Random VariableVariable whose values are uncountable

7-5


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Probability Distribution

A table, formula or graph that describes thevalues of a random variable and the probability

associated with these values

X => name of the random variable

x => a value of a random variable

P(X=x)

P(x)

7-10

Discrete ProbabilityDistributionTwo fundamental requirements

7-13

10 x P

1 all

x

x P


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Discrete ProbabilityDistribution

7-15

Number of SalesP(S)=0.2 – 3 sales calls

7-21

Call 1 Call 2 Call 3 Event x Probability

S 0.2

Sc 0.8

S 0.2

Sc 0.8

S 0.2

Sc 0.8

S 0.2

Sc 0.8

S 0.2

Sc

0.8S 0.2

Sc 0.8

S 0.2

Sc 0.8

SSS

SSSc

SScS

SSc

Sc

ScSS

ScSSc

ScScS

ScScSc

3

2

2

1

2

1

1

0


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Number of SalesP(S)=0.2 – 3 sales calls

Probability Distribution

7-23

x Probability

0

1

2

3

0.512

0.384

0.096

0.008

Describing the ProbabilityDistributionPopulation Mean

7-24

x

x xP x E

all


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Describing the ProbabilityDistributionPopulation Variance

7-26

x

x P x x V

all

22

2

all

22

x

x P x x V

Describing the ProbabilityDistributionPopulation Standard Deviation

7-27

2


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Example: Number of Pizzas

Page 223: Problem 7.20

a. Find the probability that a student has received 2 or

more pizzas this month.

7-28

X 0 1 2 3

P(X) .1 .3 .4 .2

Example: Number of Pizzas

Page 223: Problem 7.20

b. Determine the mean of the number of pizzas deliveredeach month.

7-29

X 0 1 2 3

P(X) .1 .3 .4 .2


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Laws of Variance

7-32

X V ccX V

X V c X V

cV

2

0

Binomial Distribution

Binomial Random Variable

The number of successes

in the experiment’s n trials

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Binomial Probability Tree

The probability for each sequenceof branches that represent

x successes and n-x failures has probability

To count the number of branch sequences that

produce x successes and n-x failures, we use the

combinatorial formula

7-40

x n x p p 1

!!!

x n x

nC

n

x

Binomial ProbabilityDistribution

7-41

n x

p p x n x

n x P

x n x

,,2,1,0for

1!!

!


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Binomial ProbabilityDistribution

Quiz Luck Example

10 questions, 5 choices (1 correct)

7-42

Binomial ProbabilityDistributionQuiz Luck Example

n=10, p=0.2

7-43

x P(x) P(X


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Mean and Variance ofBinomial Distribution

Mean =

Variance 2 = np 1 − p

Standard Deviationσ = np 1 − p

7-44

Poisson Distribution

Poisson Random Variable

The number of successes that occur in a period of

time or an interval of space in a Poissonexperiment.

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Poisson Experiment1. The number of successes that occur in any

interval is independent of successes that occur inany other interval

2. The probability of success in an interval is thesame for all equal-size intervals

3. The probability of a success in an interval is

proportional to the size of the interval4. The probability of > one success in an interval

approaches 0 as the interval becomes smaller

7-49


Poisson Probability Distribution

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! x

e x P

x


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No typos in a textbook

Mean of 1.5 typos per 100 pages

7-51


No typos in a 400 page textbook


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Typos in a 400 page textbook


7-53

x P(x) P(X

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OPRE 6301-SYSM 6303 Chapter 07 Slides_students