311 Session 5 2 Probability

7/29/2019 311 Session 5 2 Probability

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Probability Concepts

BUAD311 Operations Management

Session 5


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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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What is Randomness?


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Decision Making Under Uncertainty

Noahs Bagel

How many bagels do you want to bake this morning?

Netflix

How many copies ofThe Kings Speech do you

want to buy from the studio?

CBS What is the right price for a 2 minute ad during Super

Bowl 48?


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Distribution function

Every random variable is defined by its distributionfunctionF(x)

which is the probability that the outcome of the random variable

is less or equal to x

Two types of distribution functions: Discrete: it is (usually) possible to express the set of possible outcomes

as a list

e.g. the number of students that come to a given lecture: 0, 1, 2, , 50,

51, 52,

Continuous: the set of possible outcomes is unlimited and cannot be

expressed as a list

e.g. the time a sprinter takes to run the 100m dash: anything between 9

and 11 seconds (assume infinite precision)


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Discrete distribution

For each outcome we associate a

probability of occurrence

From this we can compute the

distribution function

Number of

students (x) Probability F(x)

40 0.025 0.025

41 0.05 0.025+0.05=0.03

42 0.05 0.03+0.05=0.035

43 0.05

44 0.075

45 0.1

46 0.1

47 0.15 ...

48 0.15

49 0.1

50 0.075

51 0.05 0.925+0.05=0.975

52 0.025 0.975+0.025=1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

40 41 42 43 44 45 46 47 48 49 50 51 52

Number of students

Pro

bability


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Expected value (Mean) of a discrete

distribution


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Continuous distributions

The density function is such that the area underneath corresponds to 1

(100%)

What is the probability of a particular value of occurring?

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

9.85

9.95

10.05

10.15

10.25

10.35

10.45

10.55

10.65

10.75

10.85

10.95

100%


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The normal distribution

Continuous

Defined by two parameters:

mean () and standard deviation()

Let x=54, F(x) is the probability

that the outcome is less or equal

to 54.

It is the area under the curve, on

the left of 54

What is the probability that the

outcome is greater than 54? = 48, = 6

81.13

%0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

30 33 36 39 42 45 48 51 54 57 60 63 66


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How to measure variability?

A possible measure is variance, or standard

deviation

Variance: average of the squared difference of avariable from its mean

Standard deviation: square root of the variance


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How to measure variability?


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Coefficient of Variation

A better measure of variability is the ratio of the

standard deviation to the mean. This ratio is calledthe coefficient of variation.

Coefficient of Variation = Standard Deviation (SD) /

Mean(expected value)


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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 J.Crew

The sum of throughput times at two different stagesof 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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LetXand Ybe two random variables. The sum ofX

and Y is another random variable. Let S= X+Y

The distribution ofSwill be different from that ofXandY

Example:

Let Sbe the sum of the values when you roll 2 dice

simultaneously. Let Xrepresent the value die #1

and Yrepresent the value of die #2

S= X+ Y


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The distribution of the sum S is given below:

S Prob(S) S Prob(S)

2 1/36 7 6/363 2/36 8 5/364 3/36 9 4/365 4/36 10 3/366 5/36 11 2/36

12 1/36


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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

Probability


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Expected Value and Standard Deviation of Sum of

Random Numbers

Ifa and b are known constants andXand Yare

independent random variables:

Mean[aX+bY] = a Mean[X] + b Mean[Y]

Variance[aX+bY] = a2Variance[X] + b2Variance[Y]


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Uniform Distributions

Uniform Distribution: Whenever the likelihood ofobserving 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 probability density function (pdf) for continuous U(1,5)

is f(X) = 0.25 forXbetween 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 and are

parameters of a Normal distribution)

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

Coefficient of Variation = Standard deviation / Mean

= 1


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Next Class

Waiting-line Management

How uncertainty/variability and utilization rate

determine the system performance http://www.youtube.com/watch?v=F5Ri_HhziI0&feat

ure=player_embedded

ARES reading The Psychology of Waiting-lines

A Long Line for a Shorter Wait at the Supermarket
http://www.youtube.com/watch?v=F5Ri_HhziI0&feature=player_embeddedhttp://www.youtube.com/watch?v=F5Ri_HhziI0&feature=player_embeddedhttp://www.youtube.com/watch?v=F5Ri_HhziI0&feature=player_embeddedhttp://www.youtube.com/watch?v=F5Ri_HhziI0&feature=player_embedded

311 Session 5 2 Probability

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