# Data, mode, and decisions

**Topics:**Normal distribution, Probability theory, Cumulative distribution function

**Pages:**38 (1242 words)

**Published:**September 22, 2013

Instructor’s Manual

Chapter 3

31

Manual to accompany

Data, Models & Decisions: The Fundamentals of Management Science

by Bertsimas and Freund. Copyright

2000, South-Western College Publishing. Prepared by Manuel Nunez, Chapman University.

Chapter 3

I

Chapter Outline

3.1

Continuous Random Variables

3.2

The Probability Density Function

3.3

The Cumulative Distribution Function.

The uniform distribution is introduced.

3.4

The Normal Distribution

General definition of this distribution.

Properties of normal distribution

Presentation

of

several

examples

of

normal

random

variables

and

variables

with non-normal distributions.

3.5

Computing Probabilities for the Normal Distribution

3.6

Sums of Normally Distributed Random Variables

3.7

The Central Limit Theorem

Examples of applications of this theorem.

Discussion on the normal approximation to the binomial

3.8

Summary

II

Teaching Tips

1.

The discussion on the normal approximation to the binomial and the central limit

theorem

can be enhanced by using the Crystal

Ball run option to

show in

class

how the binomial distribution approaches a bell-shaped curve as the sample size

increases. Other distributions might also be used.

2.

A quick in-class survey of the students height and weight can be used to illustrate

how

these

two

measurements

follow

normal

distributions

and

how

common

is

such distribution in nature

III

Answers to Chapter Exercises

3.1

Let X be the site of the traffic incident. X is uniformly distributed between 0 and

30 miles.

(a)

P(Travel more than 10 miles) = P(X > 10) = 1 - P(X < 10) = 1 - 10/30 = 2/3.

Instructor’s Manual

Chapter 3

32

Manual to accompany

Data, Models & Decisions: The Fundamentals of Management Science

by Bertsimas and Freund. Copyright

2000, South-Western College Publishing. Prepared by Manuel Nunez, Chapman University.

(b)

P(Travel less than 10 miles with respect to station location) = P(|X - 15| < 10)

= P(5 < X < 25) = P(X < 25) - P(X < 5) = 25/30 - 5/30 = 2/3. Hence, P(Travel

more than 10 miles with respect to station location) = P(|X - 15| > 10) = 1 -

P(|X - 15| < 10) = 1/3.

3.2

Let

T1,

T2,

and

T3

be

the

running

times

of

Donovan,

Frankie,

and

Ato,

respectively. Each of T1, T2, and T3 is uniformly distributed between 9.75 and

9.95 seconds.

(a)

P(Donovan beats record) = P(T1 < 9.86) = (9.86 - 9.75)/(9.95 - 9.75) = 0.55.

(b)

Assuming

that

the

running

times

are

independent

of

each

other,

P(Winning

time beats record) = P(min{T1, T2, T3} < 9.86) = 1

- P(min{T1, T2, T3} >

9.86) = 1 - (P(T1 > 9.86))

3

= 1 - (0.55)

3

= 0.83.

(c)

It may not be valid because runners lagging behind the leader will try to catch

up and so, their speed will depend on the speed of the leader.

3.3

Let D denote the demand for the kitchen cleanser and let d denote the inventory

level.

We

want

to

find

d

such

that

P(Stocking

out)

=

P(D

>

d)

=

0.025,

or

equivalently, P(D < d) = 0.975. Using the standard normal tables P(Z < z) = 0.975

implies z = 1.96. Therefore, d = 2,550 + (1.96)(415) = 3,363.4 bottles.

3.4

Let X denote the number of miles a pump operates before becoming ineffective.

(a)

P(Replace a pump) = P(X < 50,000) =...

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