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