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Levine Chapter 5

**RECAP **

A variable is something that varies (can take on different values)

TYPES OF VARIABLES

Type of Variable Subtype

Numerical Continuous

Can be Used in Calculations?

Yes

Categorical

Discrete Yes

No

Examples

Salary

Price

Volume of a warehouse

Income by category

Units sold

Items produced last month

Number of employees

Website visits

City

Product IDs

Satisfaction levels

WHAT TYPE OF VARIABLE DOES EACH OF THE FOLLOWING MEASURE?

What type of movie do you go to the most?

How many televisions do you have in your house?

What revenues did we get from the XYZ Company last year?

How many transactions did we have last month?

In which department do you work?

Please rate our service from 1 to 5, with 5 being excellent and 0 being poor.

What is your telephone number?

**FREQUENCY DISTRIBUTIONS VERSUS PROBABILITY DISTRIBUTIONS **

PROBABILITY IS THE L IKELIHOOD THAT A RANDOM VARIABLE WILL HAVE SOME VALUE.

Probability ranges from 0 to 1

Also expressed in percentage: 0% to 100%

SOURCES OF PROBABILITY INFORMATION

Ad hoc

Empirical

Subjective

From reasoning (cards in a deck, coin tosses)

Transaction information, surveys

Belief based on past experience

FREQUENCY DISTRIBUTIONS VERSUS PROBABILITY DISTRIBUTIONS

**Frequency Distribution**

Number of times something in a category did occur.

Revenues Last Year (Millions)

Chicago

Singapore

Praha

$491

$284

$34

**Probability Distribution**

Number of times something in a category probably will occur

Likelihood of Strike Length (Days)

0

1-5

6 or Longer

.55

.35

.10

•••They look the same but have different purposes! ••

PROBABILITY DISTRIBUTION CATEGORIES MUST BE MUTUALLY EXCLUSIVE (NO OVERLAPS)

AND COLLECTIVELY EXHAUSTIVE (TOTAL PROBABILITY = 1.0)

**Number of Classes Taken**

2

3

4

5

Total

0.24

0.16

1.0

**Probability**

0.2

0.4

**RANDOM VARIABLES **

A random variable is a variable that is *likely* to have a particular value.

EXAMPLES

If I toss a coin four times, how likely is it that I will have two heads?

If I toss two dice once, how likely is it that the total will be 7?

If customers arrive at checkout stands on average twice a minute and it takes two minutes to serve them on average, how many people will have to wait more than four minutes to be served if I have three checkout stations?

How likely is a customer to make a purchase if he or she visits our website?

How likely is it for a customer to make a purchase in the next month if they made a purchase the last time they visited our website?

COVERAGE OF RANDOM VARIABLES

DISCRETE RANDOM VARIABLES

Events must be discrete variables (whole numbers)

If I toss a coin six times, how likely is it that I will have two heads?

The number of heads is the discrete random variable.

Two heads is the event.

If I have four checkout stations and customers, customers arrive every 30 seconds on average and take two minutes to server on average, how likely will it be to have three people in line?

The number of people in line is a discrete random variable.

Three people in line is the event

**TYPES OF DISCRETE PROBABILITY DISTRIBUTIONS **

Binomial Probability Distributions

Poisson Probability Distributions

**When there is a series of trials (n) with a constant binary probability for each trial (∏). **

Example: A series of ten coin tosses. The probability of getting a head is fixed at 50% per toss. What is the probability of getting 0 heads, 1 head, 2 heads, and so forth.

Example: Rolling a single die 12 times. The probability that a 6 will be the face value is 1/6 each time. What is the probability of getting no 6s, one 6, two 6s, etc.

Example: Picking 20 light bulbs from inventory. There is a probability of 10% of picking a defective light bulb when picking each light bulb. What is the probability of getting 0 defective bulbs, 1 defective bulb, 2 defective bulbs, etc.

*Need to have a binary probability, ∏ , and the number of trials, n. *

**Number of arrivals in a given time period, given the expected number of arrivals, **

If customers come to our website on average of every 10 seconds, in a one-minute period, what is the probability that there will be one arrival, two arrivals, three arrivals, and so forth?

*Need to have the expected number of events during the time period, *

* (lambda). If visitors come an average of ten seconds and the time period is one minute, then *

* is 6. *

**BINOMIAL DISTRIBUTION **

When there is a series of trials with a constant binary probability for each trial.

Need to have a binary probability,

*∏*

, and the number of trials, *n*.

*The mean (µ) is n ∏. *

If there are ten trials and the probability of a success per trial is .20, what is the mean?

The variance (δ 2 ) is *n*

*∏(1-∏) *

If there are ten trials and the probability of a success per trial is .20, what is the variance?

EXAMPLES

IN EXCEL

**THE POISSON DISTRIBUTION **

Number of arrivals in a given time period and given an average arrival rate.

Need to have the expected number of events during the time period, (lambda). If visitors come an average of ten seconds and the time period is one minute, then

is 6.

The mean (µ) is

The variance (δ 2 ) is

.

IN EXCEL

Continue

LOOKING UP A VALUE

POISSON DISTRIBUTION SHAPE

**THIS CHAPTER **

Addressed the concept of probability distributions for random variables

Focused on discrete random variable distributions

Discussed the Binomial distribution

Discussed the Poisson distribution