Chapter 5:
Probability Distributions:
Discrete Probability Distributions
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 1
Learning Objectives

Identifying Types of Discrete Probability Distribution and their
Respective Functional Representations

Calculating the Mean and Variance of a Discrete Random
Variable with Each of the Different Discrete Probability
Distribution.
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 2
Random Variable
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 3
Random Variables

Any variable that is used to represent the outcomes any
experiment of interest to us is called Random Variable.

A random variable can assume (take) any value (Positive, negative,
zero; finite, infinite; continuous and discrete values).

Depending upon the values they take, we can identify two types of
random variables:
1. Discrete Variables
2. Continuous Variables.
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 4
Both types of variables can assume either a finite number of
values or an infinite sequence of values.
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 5
Example: JSL Appliances

Discrete random variable with a finite number of values
Let x = number of TVs sold at a given store in one day.
The number of TV units that can be sold in a given day is
finite. It is also discrete: (0, 1, 2, 3, 4).
X can be considered as Discrete Random Variable
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 6
Example: JSL Appliances

Discrete random variable with an infinite sequence of
values
Let Y = number of customers arriving in a store in one day.
Y can take on the values 0, 1, 2, . . .
We can count the customers arriving. However, there is no
finite upper limit on the number that might arrive.
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 7
Random Variables
Question
Family
size
Type
Random Variable x
X = Number of dependents
reported on tax return
Discrete
Distance from Y = Distance in miles from
home to store
home to the store site
Continuous
Own dog
or cat
Discrete
Z = 1 if own no pet;
= 2 if own dog(s) only;
= 3 if own cat(s) only;
= 4 if own dog(s) and cat(s)
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 8
Probability Distributions
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 9
Probability Distributions
The probability distribution for a random variable is a distribution
that describes the values that the random variable of interest takes.
The probability distribution is defined by a probability function,
denoted by f(x)---the probability of the values of the random
variable.
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 10
Probability Distributions
For any probability function the following conditions must be
satisfied:
1. f(x) > 0
2. f(x) = 1
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 11
Discrete Probability Distributions
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 12
Discrete Probability Distributions
A Discrete Probability Distribution is a tabular, graphic or
Functional representation of a Random Variable with discrete
outcomes that follows the principle of probability distribution
f(x) > 0
f(x) = 1
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 13
Discrete Probability Distributions--Example
• Using past data on sales, a tabular representation of the
probability distribution for TV sales was developed.
Units Sold
0
1
2
3
4
Number
of Days
80
50
40
10
20
200
x
0
1
2
3
4
f(x)
.40
.25
.20
.05
.10
1.00
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
80/200
Slide 14
A Discrete Probability Distribution can assume one or more of the
Following Distributions:
1. Uniform Distribution
2. Binomial Distribution
3. Poisson Distribution
4. Hyper-Geometric Distribution
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 16
5.1) Discrete Uniform Probability Distribution
The probability distribution of a discrete probability
distribution is given by the following formula.
f(x) = 1/n
the values of the
random variable
are equally likely
where:
n = the number of values the random
variable may assume
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 17
5.2) Binomial Distribution

The Probability distribution of a Binomial Distribution
is given by the following function
n!
x
( nx )
f (x) 
p (1  p)
x !(n  x )!
where:
f(x) = the probability of x successes in n trials
n = the number of trials
p = the probability of success on any one trial
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 18
5.3) Poisson Distribution

The Probability of a Poisson Distribution is given by
the following function
x 
 e
f ( x) 
x!
where:
f(x) = probability of x occurrences in an interval
 = mean number of occurrences in an interval
e = 2.71828
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 19
5.4) Hypergeometric Distribution

The Probability of a Hyper-geometric Distribution is
given by the following Function
 r  N  r 
 

x  n  x 

f ( x) 
N
 
n
where:
for 0 < x < r
f(x) = probability of x successes in n trials
n = number of trials
N = number of elements in the population
r = number of elements in the population
labeled success
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 20
5.1) Discrete Uniform Probability Distribution
The probability distribution of a discrete probability
distribution is given by the following formula.
f(x) = 1/n
the values of the
random variable
are equally likely
where:
n = the number of values the random
variable may assume
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 21
Expected Value (Mean) and Variance of
Discrete Probability Distributions
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 22
Expected Value (Mean) for …
…..Discrete Uniform Distribution
The expected value, or mean, of a random variable
is a measure of its central location.
E(x) =  = xf(x)
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 23
Variance and Standard Deviation of Discrete Uniform
Probability Distribution
The variance summarizes the variability in the values of a random
variable.
Var(x) =  2 = (x - )2f(x)
The standard deviation, , is defined as the square root of the
variance.
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 24
Example-TV Sales in a Given Store
• The likelihood of selling TV sets in any given day is considered equally
likely (Uniform). The following table summarizes sales data on the past
200 days.
Units Sold
0
1
2
3
4

Number
of Days
80
50
40
10
20
200
x
0
1
2
3
4
f(x)
.40
.25
.20
.05
.10
x
0
1
2
3
4
F(x)
.40
.65
.85
.90
1.00
Given this data what is the Average Number of TVs sold in a
day?
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 25
Expected Value (MEAN) and Variance

Given the data, we can find the Expected Value (Mean Number) of TVs sold
in a day as follows
x
0
1
2
3
4
E(x) =  = xf(x)
f(x)
xf(x)
.40
.00
.25
.25
.20
.40
.05
.15
.10
.40
E(x) = 1.20
expected number of
TVs sold in a day
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 26
Find the Variance and Standard Deviation of the Number of TV
Sold in a given day.
Var(x) =  2 = (x - )2f(x)
x
x-
0
1
2
3
4
-1.2
-0.2
0.8
1.8
2.8
(x - )2
f(x)
(x - )2f(x)
1.44
0.04
0.64
3.24
7.84
.40
.25
.20
.05
.10
.576
.010
.128
.162
.784
Var(x) =  2 = (x - )2f(x)= 1.66
TVs squared
Standard deviation of daily sales = 1.2884 TVs
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 27
5.2) The Binomial Distribution
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 28
5.2) Binomial Distribution

Four Properties of a Binomial Experiment
1. The experiment consists of a sequence of n
identical trials.
2. Only two outcomes, success and failure, are possible
on each trial.
3. The probability of a success, denoted by p, does
not change from trial to trial.
4. The trials are independent.
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 29
5.2) The Binomial Distribution
Typical Examples of a Binomial Experiment:
• Lottery: Win or Lose
• Election: A Candidate Wins or Loses
• Gender of an Employee: is Male or Female
• Flipping a coin: Heads or Tails
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 30
5.2) The Binomial Distribution
Our interest is in the number of successes occurring in the n trials.
let X denote the number of successes occurring in the n trials.
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 31
5.2) The Binomial Distribution

Binomial Probability Function
n!
x
( nx )
f (x) 
p (1  p)
x !(n  x )!
where:
n = the number of trials
p = the probability of success on any one trial
f(x) = the probability of x successes in n trials
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 32
5.2) The Binomial Distribution

Binomial Probability Function
n!
f (x) 
p x (1  p )( n  x )
x !(n  x )!
n!
x !(n  x )!
p x (1  p)( n  x )
Number of experimental
outcomes providing exactly
x successes in n trials
Probability of a particular
sequence of outcomes
with x successes
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 33
Binomial Distribution

Example: Evans Electronics
A local Electronics company is concerned about it
low retention of employees. In recent years,
management has seen an annual turnover of 10% in
its hourly employees. Thus, for any hourly employee
chosen at random, the company estimates that there
is 0.1 probability that the person will leave the
company in a year time.
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 34
Binomial Distribution

Given the above information, if we randomly select 3
hourly employees, what is the probability that 1 of
them will leave the company in one year ?
Solution:
Let: p = .10, n = 3, x = 1
n!
f ( x) 
p x (1  p ) (n  x )
x !( n  x )!
3!
f (1) 
(0.1)1 (0.9)2  3(.1)(.81)  .243
1!(3  1)!
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 35
Binomial Distribution

Using Tables of Binomial Probabilities
p
n
x
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
3
0
1
2
3
.8574
.1354
.0071
.0001
.7290
.2430
.0270
.0010
.6141
.3251
.0574
.0034
.5120
.3840
.0960
.0080
.4219
.4219
.1406
.0156
.3430
.4410
.1890
.0270
.2746
.4436
.2389
.0429
.2160
.4320
.2880
.0640
.1664
.4084
.3341
.0911
.1250
.3750
.3750
.1250
Page….383-388
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 37
Mean and Variance of A Binomial Distribution

Expected Value
E(x) =  = np

Variance
Var(x) =  2 = np(1  p)

Standard Deviation
  np(1  p )
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 38
Given that p=0.1, for the 3 randomly selected hourly employees,
what is the expected number and variance of workers who might
leave the company this year?

Expected Value
E(x) =  = 3(.1) = .3 employees out of 3

Variance
Var(x) =  2 = 3(.1)(.9) = .27

Standard Deviation
  3(.1)(.9)  .52 employees
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 39
5.3) Poisson Distribution
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 40
6.4. Poisson Distribution

Poisson distribution refers to the probability distribution of a
trial that involves cases of rare events that occur over a fixed
time interval or within a specified region
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 41
6.4. Poisson Distribution
Examples….
• The number of errors a typist makes per page
• The number of cars entering a service station per hour
• The number of telephone calls received by a switchboard per hour.
• The number of Bank Failures During a given Economic Recession.
• The number of housing foreclosures in a given city during a given year.
• The number of car accidents in one day on I-35 stretch from the Twin
Cities to Duluth
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 42
Poisson Distribution
We use a Poisson distribution to estimate the number of
occurrences of such discrete incidents over a specified
interval of time or space
Thus a Poisson distributed random variable is discrete;
Often times it assumes an infinite sequence of values
(x = 0, 1, 2, . . . ).
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 43
Properties of a Poisson Experiment

The number of successes (events) that occur in a certain time interval is
independent of the number of successes that occur in another time interval.

The probability of a success in a certain time interval is
• the same for all time intervals of the same size,
• proportional to the length of the interval.

The probability that two or more successes will occur in an interval
approaches zero as the interval becomes smaller.
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 44
Poisson Distribution

Poisson Probability Function
x 
 e
f ( x) 
x!
where:
f(x) = probability of x occurrences in an interval
 = mean number of occurrences in an interval
e = 2.71828
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 45
Poisson Distribution--Example
On Average 6 Patients per hour
arrive at the emergency room
of Mercy Hospital on
weekend evenings.
MERCY
What is the probability of 4 arrivals in
30 minutes on a weekend evening?
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 46
Poisson Distribution-Example

MERCY
Using the Poisson Probability Function
f ( x) 
x 
 e
x!
= 6/hour = 3/half-hour, P(x = 4)?
3 4 (2.71828)3
f (4) 

4!
.1680
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 47
MERCY
Poisson Distribution

Using Poisson Probability Tables

x
0
1
2
3
4
5
6
7
8
2.1
.1225
.2572
.2700
.1890
.0992
.0417
.0146
.0044
.0011
2.2
.1108
.2438
.2681
.1966
.1082
.0476
.0174
.0055
.0015
2.3
.1003
.2306
.2652
.2033
.1169
.0538
.0206
.0068
.0019
2.4
.0907
.2177
.2613
.2090
.1254
.0602
.0241
.0083
.0025
2.5
.0821
.2052
.2565
.2138
.1336
..0668
.0278
.0099
.0031
2.6
.0743
.1931
.2510
.2176
.1414
.0735
.0319
.0118
.0038
2.7
.0672
.1815
.2450
.2205
.1488
.0804
.0362
.0139
.0047
2.8
.0608
.1703
.2384
.2225
.1557
.0872
.0407
.0163
.0057
Page---390-395
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
2.9
.0550
.1596
.2314
.2237
.1622
.0940
.0455
.0188
.0068
3.0
.0498
.1494
.2240
.2240
.1680
.1008
.0504
.0216
.0081
Slide 48
Mean and Variance of Poisson Distribution
Another special property of the Poisson distribution
is that the mean and variance are equal.
=2
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 50
Poisson Distribution

MERCY
Variance for Number of Arrivals
During 30-Minute Periods
=2=3
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 51
5.4) Hyper-Geometric Distribution
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 52
Hyper-geometric Distribution
The hyper-geometric distribution is closely related
to the binomial distribution.
However, for the hyper-geometric distribution:
the trials are not independent, and
the probability of success changes from trial
to trial.
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 53
Hyper-geometric Distribution

Hyper-geometric Probability Function
 r  N  r 
 

x  n  x 

f ( x) 
N
 
n
where:
for 0 < x < r
f(x) = probability of x successes in n trials
n = number of trials
N = number of elements in the population
r = number of elements labeled as success
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 54
Hyper-geometric Distribution

Hyper-geometric Probability Function
f (x) 
r
x
 
N r
nx


N
n
 
number of ways
x successes can be selected
from a total of r successes
in the population
for 0 < x < r
number of ways
n – x failures can be selected
from a total of N – r failures
in the population
number of ways
a sample of size n can be selected
from a population of size N
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 55
Hyper-geometric Distribution

Mean
 r 
E ( x)    n  
N

Variance
r  N  n 
 r 
Var ( x)    n  1  

 N  N  N  1 
2
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 56
Hyper-geometric Distribution: Inspection
Electric fuses produced by a given company are
packed in boxes. Each box carries 12 units of
electric fuses. The role of the inspector in
the company that manufactures the fuses is to
make sure that all fuses in each box are in
good condition.
Consider the following scenario: A worker
inadvertently places 5 defective items in a
box. As part of her job, the inspector
randomly selects 3 fuses from the box that
contains the defective fuses.
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 57
Hyper-geometric Distribution: Inspection
1.
What is the probability that none of the three
randomly selected fuses are defective?
2.
What is the probability that the inspector
finds only one of the three randomly selected
fuses to be defective?
3.
What is the probability that the inspector
finds at least one of the three fuses
defective?
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 58
Hyper-geometric Distribution: Inspection
What is the probability that none of the
three randomly selected fuses are
defective?
Solution:
N=12; r=5; n=3; x=0; P(x=0)?
1.
 5!  7! 



5 125
( 0 )( 30 )  0!5!  3!4!  35
f ( x  0)  12


 0.1591
(3 )
220
 12! 


 3!9! 
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 59
Hyper-geometric Distribution: Inspection
2. What is the probability that the inspector
finds only one of the three randomly
selected fuses to be defective?
Solution:
N=12;
r=5;
n=3;
x=1;
P(x=1)?
 5!  7! 
 

5 125
(1 )( 31 )  1!4!  2!5!  5 X 21
f ( x  1)  12


 0.4773
(3 )
220
 12! 


 3!9! 
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 60
Hyper-geometric Distribution: Inspection
2. What is the probability that the inspector
finds at least one of the three fuses
defective?
Solution:
N=12;
r=5;
n=3;
x

1
f ( x  1)  ?
 1  p( X  0)
 1  0.1591
 0.8409
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 61
Hyper-geometric Distribution

Example: Neveready
Bob has removed two dead batteries from a
flashlight and inadvertently mingled them with the
two good batteries that he intended to use as
replacements. The four batteries look identical.
Then Bob randomly selects two of the four batteries.
What is the probability that he selects the two good
batteries?
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 62
Hyper-geometric Distribution

Using the Hyper-geometric Function
2!  2!2! 
 r  N  r   2  2   2!
 
 x  n  x   2  0  2!0! 
2!0!0!2!
0!2! 1 1








f (x) 


  .167
0.167
N
4
4!
66
 
 
 4! 
n
2
2!2! 
 
 


 2!2! 
N = 4 = number of batteries in total
r = 2 = number of good batteries in total
x = 2 =number of good batteries to be selected
n = 2 = number of batteries selected
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 63
Hyper-geometric Distribution
When the population size is large, a hyper-geometric
distribution can be approximated by a binomial
distribution with n trials and a probability of success
p = (r/N).
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 66
Hyper-geometric Distribution
Consider a hyper-geometric distribution with n trials
and let p = (r/n) denote the probability of a success
on the first trial.
If the population size is large, the term (N – n)/(N – 1)
approaches 1.
The expected value and variance can be written
E(x) = np and Var(x) = np(1 – p).
Note that these are the expressions for the expected
value and variance of a binomial distribution.
2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse)
Slide 67