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8/17/2019 OPRE 6301-SYSM 6303 Chapter 07 Slides_students
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9/23/20
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
8/17/2019 OPRE 6301-SYSM 6303 Chapter 07 Slides_students
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8/17/2019 OPRE 6301-SYSM 6303 Chapter 07 Slides_students
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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.
7-45
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Poisson Distribution
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 Distribution
Poisson Probability Distribution
7-50
! x
e x P
x
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Poisson Distribution
No typos in a textbook
Mean of 1.5 typos per 100 pages
7-51
Poisson Distribution
No typos in a 400 page textbook
Mean of 1.5 typos per 100 pages
7-52
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Poisson Distribution
Typos in a 400 page textbook
Mean of 1.5 typos per 100 pages
7-53
x P(x) P(X