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Operations Management
Session 10: Probability Concepts
Simulation Game
 Game codes due.

Please go to http://usc.responsive.net/lt/usc/start.html to
register.

Course code: usc.

Individual code: what you purchased from the bookstore.
 Case groups posted. Please double-check.
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Today’s Class
 Probability Concept Review
 Basic Statistics Formula
 Common Distribution
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Quote of the day
 Without the element of uncertainty, the bringing
off of even, the greatest business triumph would
be dull, routine, and eminently unsatisfying.
J. Paul Getty
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Blackjack
 You have a 9 and 5, what will happen if you hit?
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Random Experiment
 Random Experiment: An experiment in which the precise
outcome is not known ahead of time. The set of possibilities
however is known
 Examples:

Demand for blue blazers next month
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The value of a rolled die
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The waiting times of customers in the bank
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The waiting time for an ATT service person
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Tomorrow’s closing value of the NASDAQ
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The temperature in Los Angeles tomorrow
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Random Variable
 A random variable is the numerical value determined by the
outcome of a random experiment
 A random variable can be discrete (i.e. takes on only a
finite set of values) or continuous

Examples:

The value on a rolled die is a discrete random variable

The demand for blazers is a discrete random variable

The birth weight of a newborn baby is a continuous variable

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The waiting time for the AT&T service person is a continuous
random variable
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Sample Space
 Sample space is the list of possible outcomes of an
experiment
 Examples:
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

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For a die, the sample space S is: {1,2,3,4,5,6}
For the demand for blue blazers it is all possible realizations of the
demand. For example: {1000,1001,1002…,2000}
The waiting time in the bank is any number greater than or equal to
0. This is a continuous random variable
The waiting time for a bus at a bus stop is any number between 0
and 30 minutes. This is a continuous random variable that is
bounded
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Event
 An event is a set of one or more outcomes of a
random experiment
 Examples:


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Getting less than 5 by rolling the die: This event occurs if
the values observed are {1,2,3, or 4}
The demand is smaller or equal to 1500. This event occurs
if the values of the demand are {1000, 1001, … 1500}
The waiting time for a bus at the bus stop exceeds 10.
This event occurs if the wait time is in the interval (10, 30)
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Probability
 The probability of an event is a number between 0
and 1
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

1 means that the event will always happen
0 means that the event will never happen
The probability of an event A is denoted as either P(A) or
Prob(A)
 Example: Probability of Rolling die and observing a
number less than 5 =
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P(outcome< 5) = Prob(observing {1,2,3 or 4}) = 4/6 = 2/3
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Probability
 Probability that A doesn’t occur:
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P(not A) = 1 – P(A)
Thus, the probability you will roll a number larger or equal to
5 is or 6 is: 1 – Probability (Outcome <5) = 1 – 2/3 = 1/3
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Probability
 Suppose all the outcomes that constitute the “waiting time”
for an AT&T operator are equally likely. The minimum
waiting time is 30 min and the maximum is 90 min.
 Then the probability of waiting less than 45 min is:
Event
Sample Space
 P (waiting more than 45 min) is:
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45  30
 0.25
90  30
90  45
 0.75
90  30
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Probability Distribution for
Discrete Random Variables
 Let us begin with discrete outcomes
 A probability distribution is a list of:

All possible values for a random variable (Sample space); and

The corresponding probabilities
 For a die,
the probability distribution is:
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Outcome
Probability
1
1/6
2
1/6
3
1/6
4
1/6
5
1/6
6
1/6
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Probability Distribution for
Discrete Random Variables
 The chart below depicts the probability distribution
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Cumulative Probability Distribution
for Discrete Random Variables
 Probability that a random number will be less than or
equal to some given number
 For a die, the cumulative
probability distribution is:
Outcome less
than or equal
to:
Additional: What is the
probability a die roll is
less than 3.5?
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Probability
1
1/6
2
2/6
3
3/6
4
4/6
5
5/6
6
1
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Continuous Random Variables and
Probability Density Functions (PDF)
 The probability density function is the analog of the
probability distribution (table 1) for discrete random
numbers
 Example: Suppose we have a computer program that can
generate any number between 1 and 6 (not just the
integers)


Assume that each number is equally likely to be generated.
Then we have a continuous random number
This random number has a uniform distribution between 1
and 6
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Continuous Random Variables and
Probability Density Functions (PDF)
Probability Density
1/5
1
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2
3 4 5
Outcome
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Properties of Probability Density
Functions

By convention the total area under the probability density
function must equal 1
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
The base of the rectangle in the figure is 6 – 1 = 5 units long, the
probability density is 1/5 for all values between 1 and 6. This
ensures that the total area is 1
The probability of observing any value between two
numbers is equal to the area under the probability density
function between those numbers

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The probability of observing any number between 4.0 and 5.0 will
be (5.0 – 4.0)* 1/5 = 1/5 = 0.2
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Properties of Probability Density
Functions
Probability Density, f
1/5
1
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3
4
5
Outcome
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Properties of Cumulative
Distribution Functions
CDF, F
Probability that the
outcome is smaller
than 5: is 4/5
Probability that the
outcome is smaller
than 2: is 1/5
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1
4/5
1/5
0
1
2
3 4 5
Outcome
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Relationship between Density and
Cumulative Distribution Functions
CDF
1
4/5
1/5
0
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1
2
3 4 5
Outcome
6
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Other distributions: Triangular
Probability that the outcome
is between 2 and 5
2
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Normal Distribution
0.80
0.70
0.60
Normal distribution #2
Normal distribution #1
0.50
0.40
0.30
0.20
0.10
0.00
-4
-3
-2
-1
0
1
2
3
4
X
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Continuous Random Variables and
Probability Density Functions (PDF)
0.80
0.70
0.60
Normal distribution #2
Normal distribution #1
0.50
0.40
0.30
0.20
0.10
0.00
-4
-3
-2
-1
0
1
2
3
4
X
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Cumulative Density Function (CDF)
for Continuous Random Numbers
 This is analogous to the cumulative distribution
function for discrete random numbers
 The cumulative density function gives the
probability of the continuous random variable
being equal to or smaller than a given number
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Cumulative Density Function (CDF)
for Continuous Random Numbers
Cumulative
Density 1
Function
1/5
o
1
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3
4
5
6
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Mean or Expected Value of a
Random Number
 Expected value can be thought of as the average value
of a random variable
 Let us denote by X the value of the random variable.
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If the random variable is the value of a die, then X denotes
the value rolled. If we roll a 6, then X = 6). We will use the
notation E[X] to denote the expected value of X
 If the random number is a discrete variable that can
take on values between 1 and N then:

E[X] = Thus for the die, E[X] = 1/6*1 + 1/6*2 + 1/6*3 +
1/6*4 + 1/6*5 + 1/6*6 = 3.5
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Mean or Expected Value of a
Random Number
 What if the variable is a continuous random variable?

Let f(X) be the probability density function.
 Example: for the uniform distribution, we have seen:

f(X) = 0.2 whenever X is between 1 and 6. f(X) = 0 if X is not
between 1 and 6.]
6
E ( X )   xf ( x)dx
1
 Integration of continuous variables in lay terms is
equivalent to summation for discrete variables.
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The Variance of X
 When X is a discrete random variable:
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Var(X) =  (X – E[X])2*Prob(X)
 If X is the random number generated by the roll of
a die then:
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
Var(X) = (1-3.5)2*1/6 + (2-3.5)2*1/6 +(3-3.5)2*1/6 +(43.5)2*1/6 +(5-3.5)2*1/6 +(6-3.5)2*1/6 = 2.9166
Standard Deviation = square root of variance
SD(X) = 1.708 in this example
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How to measure variability?
 A possible measure is variance, or standard
deviation
 Is this good enough?
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195
180
165
150
135
120
105
90
75
60
45
0
30
0
15
0.005
38.4
0.01
36
0.01
33.6
0.02
31.2
0.015
28.8
0.03
26.4
0.02
24
0.04
21.6
0.025
19.2
0.05
16.8
0.03
14.4
0.06
12
0.035
9.6
0.07
7.2
0.04
4.8
0.08
2.4
0.045
0
0.09
0
Which one has the larger
variability?
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Which one has the larger
variability?
 The variation in the first set appears to be
significantly higher than the second set.
 Nevertheless, the standard deviation of the first
graph is 5, the standard deviation of the second
graph is 10.
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Coefficient of Variation
 A better measure of variability is the ratio of the
standard deviation to the average. This ratio is
called the coefficient of variation.

Coefficient of Variation = Standard Deviation / Average (expected
value)
 A similar measure is squared coefficient of variation:
SCV = (CV)2 = (SD/M)2
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Sum of Random Numbers
 Often we have to analyze sum of random numbers.
 Examples include:
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The sum of the demand of different products processed by
the same resource
 The total demand for cars produced by GM

The total demand for knitwear at DD
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The total completion time of a project
The sum of throughput times at two different stages of a
service system (waiting time to place an order at a cafeteria
and waiting time in the line to pay for the food)
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Sum of Random Numbers
 Let X and Y be two random variables. The sum of X
and Y is another random variable. Let S = X +Y
 The distribution of S will be different from that of X
and Y
 Example:


Let S be the sum of the values when you roll 2 dice
simultaneously. Let X represent the value die #1 and Y
represent the value of die #2
S=X+Y
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Sum of Random Numbers
 The distribution of the sum S is given below:
S
Prob(S)
S
Prob(S)
2
1/36
7
6/36
3
2/36
8
5/36
4
3/36
9
4/36
5
4/36
10
3/36
6
5/36
11
2/36
12
1/36
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Sum of Random Numbers
 E[S] = 2*1/36 + 3*2/36 + 4*3/36 + 5*4/36 +
6*5/36 + 7*6/36 + 8*5/36 + 9*4/36 + 10*3/36 +
11*2/36 + 12*1/36 = 7
 Var(S) = (2 - 7)2*1/36 + (3 - 7)2*2/36 +…….+ (12 7)2*1/36 = 5.83
 SD(S) = 5.83^1/2 = 2.42
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Sum of Random Numbers
0.18
0.16
Probability
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
2
3
4
5
6
7
8
9
10
11
12
Sum of the two rolls
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Expected Value and Standard Deviation
of Sum of Random Numbers
 If a and b are 2 known constant and X and Y are
random independent variables:
 E[aX+bY] = aE[X] + bE[Y]
 Var(aX+bY) = a2Var(X) + b2Var(Y)
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Specific Distributions Of Interest
 We will also utilize Uniform Distributions
 Uniform Distribution: Whenever the likelihood of
observing a set of numbers is equally likely
 Continuous or discrete
 We use notation U(a,b) to denote a uniform
distribution



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Example U(1,5) is uniform distribution between 1 and 5.
If it is a discrete distribution then outcomes 1,2,3,4, and 5
are equally likely (each with probability 1/5)
If it is a continuous distribution then all numbers between
1 and 5 are equally likely
The p.d.f. for U(1,5) (continuous) will be f(X) = 0.25 for X
between 1 and 5
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Exponential Distribution
 The exponential distribution is often used as a
model for the distribution of time until the next
arrival.


The probability density function for an Exponential distribution
is: f(x) = e-x, x > 0
 is a parameter of the model (just as m and s are parameters
of a Normal distribution)

E[X] = 1/

Coefficient of Variation = Standard deviation / Average = 1
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Var(X) = 1/2
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Exponential Distribution
Shape of the Exponential Probability Density Function
f(X)
X
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Poisson Distribution
 The Poisson Distribution is often used as a model for
the number of events (such as the number of
telephone calls at a business or the number of
accidents at an intersection) in a specific time period

The probability of n events is: p(n) = ne-/n!, n = 0, 1, 2, 3,
…

 is a parameter of the model

E[N] = 

Var(N) = 
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Poisson Distribution
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Next Class
 Waiting-line Management

How uncertainty/variability and utilization rate determines the
system performance
 Article Reading: “The Psychology of Waiting-lines”
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