1

Business Statistics

Lecture 3: Random Variables and the

Normal Distribution

2

• A little bit of probability

• Random variables

• The normal distribution

Goals for this Lecture

3

Probability vs. Statistics

• Probability: You assume a mechanism that generates particular outcomes and then calculate the chance of other outcomes• E.g., Given a “fair” coin, what is the chance of

flipping four heads and two tails out of six flips?

• Statistics: After seeing some outcomes, you try to say something about the mechanism generating the outcomes• E.g., After flipping four heads and two tails you

ask, “What are the chances this coin is fair?”

• Two sides of the same coin (pun intended!)• Language of probability common to both

4

Definitions

• Sample space (S)

• Set of all possible

outcomes of an experiment

• Event

• Collection of one or more

outcomes

• Probability

• Function assigning a

number from 0 to 1 to

events, subject to rules

S

AS

Venn Diagrams

5

Examples:

Sample Spaces and Events

• Roll a fair die: S = {1,2,3,4,5,6}

• Simple events

• Roll is a 1

• Roll is a 6

• Compound event

• Roll is even: {2, 4, 6}

• Roll is less than 4: {1, 2, 3}

• Fair: each simple event is equally likely

• Other sample space examples:

• Flip of one coin: S = {H, T}

• Flip two coins: S = {(H,H), (H,T), (T,H), (T,T)}

6

Set Theory Terminology

• Union: A B• outcomes in event A or event B or both

• Intersection: A B• outcomes in both event A and event B

• Complement: Ac

• outcomes in S not in event A

• Mutually exclusive or disjoint events• events with no outcomes in common

S

A B

S

AB

S

A B

S

A

Venn Diagrams

Ac

7

Some Notation

• Pr(A) or P(A) is shorthand for “the probability

that event „A‟ occurs”

• For a coin, we might write Pr(H) to mean the

probability that a head occurs, for example

• If we define N(A) as the number of “A” events

in the sample space, then

under the assumption that all simple events

are equally likely

S

HNH

in outcomes ofnumber total

)()Pr(

2

1

8

• Disjoint (or mutually exclusive) events: Both

events cannot happen at the same time

• Either A or B (or “not A or B”) will happen

• Probability of the union of two disjoint events:

Pr(A or B) = Pr(A U B) = Pr(A) + Pr(B)

AB

• Ex: Probability of rolling a 1 or a 2 on a die:– Pr(roll 1 or 2) = Pr(roll 1) + Pr(roll 2) = 1/6 + 1/6 = 1/3

Probability of the

Union of Disjoint Events

9

General Rule for Probability of

the Union of Two Events

• Both events can happen at the same time

• Yellow/green striped region

• In general, probability of the union of two events:

Pr (A and b) = Pr(A U B) = Pr(A) + Pr(B) – Pr(A B)

• Pr(A B) is the intersection of A and B

• Basically, the striped area is counted twice in

Pr(A) + Pr(B), so one must be subtracted off

• When events are disjoint Pr(A B) = 0

AB

U

U

U

10

• Complementary events: Either one or

the other will happen, but not both

• Either A will happen or “not A” will happen

• Pr(not A) = Pr(Ac) = 1 - Pr(A)

ANot A

• Ex: The probability that you do not roll a 3 is

1 minus the probability that you roll a 3– Pr(not roll 3) = 1 – Pr(roll 3) = 1 – 1/6 = 5/6

Probability An Event

Will Not Happen

Probability of the Intersection of

Independent Events

• Independent events:

Two observations are

independent if knowing

the value of one doesn‟t help you guess the

value of the other

• Rule: Pr(A and B) = Pr(A B) = Pr(A) x Pr(B)

• Example: In two rolls, the probability you roll

a 1 both times

• Pr( roll a 1 both times)

= Pr(roll 1 on first roll) x Pr(roll 1 on second roll)

= 1/6 x 1/6 = 1/3611

AB

U

12

Dependence

• Opposite of independence

• Knowing the value of one observation

helps you guess the value of another

• Example: The average price of GM‟s stock

was $59.50 in September. What will the

average price be for October?

• Your best guess uses the September

information, so the average monthly stock

prices are dependent

13

Variables vs. Random Variables

• A variable is simply a notational placeholder for a measured or observed value• E.g., let the variable A equal your age

• For me, A=47 (years)

• A random variable is a variable for a random observation• E.g., let the random variable X be the age of a

random person in the class

• For a specific person, X has a value

• For a collection of people, X has a distribution, which gives the frequency of occurrence of ages in class

14

Random Variables

• From the first class:

Let X be the outcome of a dice roll

• X is a random variable

• X can be equal to 1, 2, 3, 4, 5, or 6

depending on what occurs on the roll of a

dice

• X has a distribution:

• Probability X=x is 1/6, for x=1,2,3,4,5, or 6

• Notation: Pr(X=x)=1/6, for x=1,2,3,4,5, or 6

15

Plotting a Probability Distribution

• Let X denote the outcome of a fair die

• i.e., Pr(X=x)=1/6, for x=1,2,3,4,5, or 6

• We can draw the probability function:

Pr

(X=

x)

x

1/6

2/6

0 1 2 3 4 5 6

0

16

Probability Distributions

• Can be for either discrete or continuous

variables (data)

• Gives the probability of an event or set

of events

• Sum over all possible events equals 1

• Means one of the possible events must

happen

• E.g., Rolled die must give a 1, 2, 3, 4, 5,

or 6

17

• Normal Distribution is an important

continuous distribution

• Symmetric, bell-shaped

• For population, described by its

• Mean:

• Standard deviation:

• Notation: N( , 2)

• Being non-normal does not mean

abnormal

Normal Distributions

Greek letter “mu”

Greek letter “sigma”

18

Properties of the Normal Curve

• Symmetric

• Bell shaped

• Unimodal

• “Thin tails”

• The normal curve is a model relating the

mean and variance to the quantiles

19

Why Focus on the

Normal Distribution?

• Normal distribution describes many

natural phenomenon well

• Central Limit Theorem explains why

• Statistical theorem: Distribution of sums of

random variables tends toward the normal

• The more things that are summed, the

more like the normal

• Result is that averages tend to have a

normal distribution

Central Limit Theorem in Action

.01

.05

.10

.25

.50

.75

.90

.95

.99

-3

-1

0

2

.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

Hig

hly

skew

ed

pro

cess

.0 .1 .2 .3 .4 .5 .6 .7

.01

.05

.10

.25

.50

.75

.90

.95

.99

-3

-1

1

3

Mean of 5

•.01

•.05

•.10

•.25

•.50

•.75

•.90•.95

•.99

-3

-2

-1

0

1

2

3

.0 .1 .2 .3 .4 .5 .6

Mean of 10 .01

.05

.10

.25

.50

.75

.90

.95

.99

-3

-2

-1

0

1

2

3

.2 .3

Mean of 100

21

• A statistic is a one-number summary of data

• Statistics can be for samples or populations

• x-bar and s are examples of sample statistics

• and are parameters of the normal

distribution

• We often estimate parameters with statistics

• Estimate with

• Estimate with s

Statistics and Parameters

X

22

…has a corresponding

sample summary

• Mean

• Standard deviation

• Proportion

Statistics

x

s

p̂

Sample statistics are good guesses for

population parameters, but they’re not the

same

Parameters vs. Statistics

• Every population

summary…

• Mean ()

• Standard Deviation

()

• Proportion (p)

Parameters

23

The Empirical Rule

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

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

Z

• If the normal curve

fits well then:

99%

• 99% within 3 SD68%

• 68% of the data is

within 1 SD of the

mean

95%

• 95% within 2 SD

24

“Standardizing” a

Normal Distribution

• Standardizing means turning an

observation from a N( , 2) into a

N(0,1) observation

• If X comes from a N( , 2) then

has a N(0,1) distribution

• If and are estimated, then use

XZ

X x

Zs

25

• See Table A3-2 on page 491 in Business Statistics

• Enter with a value of “a”

• Read across to the “p” column to get probability of being between a and –a

• Example: a=1• Probability is 0.6827 of being between 1 and –1

• Empirical rule!

• Note: Can also go in the other direction to find the a-value corresponding to a probability

Finding the Probability

for a Normal Distribution (1)

26

• See Table A3-1 on page 492 in Business Statistics

• Enter with a value of “a”

• Read across to the “p” column to get probability of being less than a

• Example: a=1: Probability is 0.8413

• Example: a=-1: Probability is 0.1587

• So, Pr(-1<z<1) = Pr(z < 1) – Pr(z < -1)

=0.8413 – 0.1587 = 0.6826• Empirical rule again!

Finding the Probability

for a Normal Distribution (2)

27

• In Excel, use NORMSDIST function

• “=NORMSDIST(a)” = Pr(Z<a)

• Just like Table A3-1

• Can also use NORMDIST function

• Gives probability for any normal distribution

• Form: =NORMDIST(a,,,1)• So, NORMDIST(a,0,1,1) = NORMSDIST(a)

Finding the Probability

for a Normal Distribution (3)

28

• For a standard normal distribution:

• What is the probability of being outside of the interval (3, -3)?

• What is the probability of getting an observation less than –2?

• For a N(1,32) distribution:

• What is the probability of being within one standard deviation of the mean?

• What is the probability of getting an observation greater than 7?

Exercises in finding the Probability

for a Normal Distribution

Solutions…

29

30

Using Normal Probabilities

to Test Assertions

• You are production manager of a widget

manufacturing facility

• Defective widget: quality characteristic < 7

• Line supervisor says not to worry:

• Distribution of quality characteristic is N(16,9)

• Should you worry or not?

• You‟re a careful production manager

• Visit the line and pick a random widget

• Widget‟s quality characteristic measures 5

• Do you believe the supervisor‟s distribution

assertion?

The Calculations…

31

32

Testing Normality

• Normal Quantile Plot, or “Q-Q” Plot

• X-axis: observed data

• Y-axis: expected data if normal model were true

• Close to straight line means close to normal

• JMP: After Analyze > Distribution > red triangle >

Normal Quantile Pl

400

500

600

700

800.01 .05.10 .25 .50 .75 .90.95 .99

-2 -1 0 1 2 3

Normal Quantile Plot

GMAT Case

-0.07

-0.05

-0.03

-0.01

0.01

0.03

0.05

0.07

50 100 150 200

Count Axis

.01 .05.10 .25 .50 .75 .90.95 .99

-2 -1 0 1 2 3

Normal Quantile Plot

GM Stock Case

Rel

Chan

ge

GM

AT

Sco

re

Evaluating Normality

.01

.05

.10

.25

.50

.75

.90

.95

.99

-3

-1

0

2

.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

Hig

hly

skew

ed

pro

cess

.0 .1 .2 .3 .4 .5 .6 .7

.01

.05

.10

.25

.50

.75

.90

.95

.99

-3

-1

1

3

Mean of 5

•.01

•.05

•.10

•.25

•.50

•.75

•.90•.95

•.99

-3

-2

-1

0

1

2

3

.0 .1 .2 .3 .4 .5 .6

Mean of 10 .01

.05

.10

.25

.50

.75

.90

.95

.99

-3

-2

-1

0

1

2

3

.2 .3

Mean of 100

In the Readings…

• …don‟t worry too much about:

• Sampling with and without replacement

• Permutations and combinations

• Uniform, t, chi-square, and F distributions

• If we had more time, we‟d cover these

topics

34

35

What we have learned so far…

• Types of data and types of variation

• Descriptive statistics

• Statistical plots and graphs

• Random variables and a little probability

• “And,” “Or,” “Not” rules

• The normal distribution

• Standardizing

• Calculating probabilities