Operations Management Session 10: Probability Concepts.

Operations Management

Session 10: Probability Concepts

Session 10 Operations Management 2

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.


Today’s Class

Probability Concept Review

Basic Statistics Formula

Common Distribution


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


Blackjack

You have a 9 and 5, what will happen if you hit?


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 The value of a rolled die The waiting times of customers in the bank The waiting time for an ATT service person Tomorrow’s closing value of the NASDAQ The temperature in Los Angeles tomorrow


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 The waiting time for the AT&T service person is a

continuous random variable


Sample Space

Sample space is the list of possible outcomes of an experiment

Examples: 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


Event

An event is a set of one or more outcomes of a random experiment

Examples: 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)


Probability

The probability of an event is a number between 0 and 1

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 =

P(outcome< 5) = Prob(observing {1,2,3 or 4}) = 4/6 = 2/3


Probability

Probability that A doesn’t occur: 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


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:

P (waiting more than 45 min) is:

25.03090

3045

Event

Sample Space

75.03090

4590


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:

Outcome Probability

1 1/6

2 1/6

3 1/6

4 1/6

5 1/6

6 1/6


Probability Distribution for Discrete Random Variables

The chart below depicts the probability distribution


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: Probability

1 1/6

2 2/6

3 3/6

4 4/6

5 5/6

6 1

Additional: What is the probability a die roll is less than 3.5?


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



Functions (PDF)

Probability Density

Outcome

1 2 3 4 5 6

1/5


Properties of Probability Density Functions

By convention the total area under the probability density function must equal 1

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

The probability of observing any number between 4.0 and 5.0 will be (5.0 – 4.0)* 1/5 = 1/5 = 0.2


Properties of Probability Density Functions

Probability Density, f

Outcome1 2 3 4 5 6

1/5


Properties of Cumulative Distribution Functions

CDF, F

Outcome

1 2 3 4 5 60

1

1/5Probability that the outcome is smaller than 2: is 1/5

4/5

Probability that the outcome is smaller than 5: is 4/5


Relationship between Density and Cumulative Distribution

Functions

CDF

Outcome

1 2 3 4 5 60

1

1/5

4/5


Other distributions: Triangular

2 5

Probability that the outcome is between 2 and 5


Normal Distribution

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

-4 -3 -2 -1 0 1 2 3 4

X

Normal distribution #1 Normal distribution #2



Functions (PDF)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

-4 -3 -2 -1 0 1 2 3 4

X

Normal distribution #1 Normal distribution #2


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


Cumulative Density Function (CDF) for Continuous Random

Numbers

1 2 3 4 5 6

CumulativeDensityFunction

o

1

1/5


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.

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


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.]

Integration of continuous variables in lay terms is equivalent to summation for discrete variables.

6

1

)()( dxxxfXE


The Variance of X

When X is a discrete random variable: Var(X) = (X – E[X])2*Prob(X)

If X is the random number generated by the roll of a die then:

Var(X) = (1-3.5)2*1/6 + (2-3.5)2*1/6 +(3-3.5)2*1/6 +(4-3.5)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


How to measure variability?

A possible measure is variance, or standard deviation

Is this good enough?


Which one has the larger variability?

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

02.

44.

87.

29.

6 1214

.416

.819

.221

.6 2426

.428

.831

.233

.6 3638

.4

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0 15 30 45 60 75 90 105

120

135

150

165

180

195


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.


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


Sum of Random Numbers

Often we have to analyze sum of random numbers. Examples include:

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 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)



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



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



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



0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

2 3 4 5 6 7 8 9 10 11 12

Sum of the two rolls

Pro

babi

lity


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)


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 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


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 and are parameters of a Normal distribution)

E[X] = 1/ Var(X) = 1/2

Coefficient of Variation = Standard deviation / Average = 1


Exponential Distribution

Shape of the Exponential Probability Density Function

f(X)

X


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) =


Poisson Distribution


Next Class

Waiting-line Management How uncertainty/variability and utilization rate

determines the system performance

Article Reading: “The Psychology of Waiting-lines”

Operations Management Session 10: Probability Concepts.

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Transcript of Operations Management Session 10: Probability Concepts.