IS 310
Business
Statistics
CSU
Long Beach
IS 310 – Business Statistics
Slide 1
Chapter 5
Discrete Probability Distributions
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Random Variables
Discrete Probability Distributions
Expected Value and Variance
Binomial Probability Distribution
Poisson Probability Distribution
.40
.30
.20
.10
0
IS 310 – Business Statistics
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2
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Slide 2
Random Variables
A random variable is a variable that can take on values
at random. Consider the following experiments:
1. Asking 10 students if they watched a TV show last
night (the number of students who watched the
show is a random variable)
2. Inspecting 20 items of a product to check quality of
the items (the number of defective items is a random
variable)
3. Tossing a coin five times (the number of heads
occurring is a random variable)
4. Taking an exam with 100 questions (the number of
correct answers is a random variable)
IS 310 – Business Statistics
Slide 3
Random Variables (Contd)
A random variable can be either
Discrete or Continuous
Discrete random variables take on certain specific values.
Examples are the following: number of defective items in an
inspection (0, 1, 2, 3,….); number of correct answers in an exam
(0, 1, 2, 3, …); number of heads obtained in tossing a coin five
times (0, 1, 2, 3, 4, 5)
o---------o---------o---------o---------o
The only values the discrete random variable can take on are
indicated by circles
IS 310 – Business Statistics
Slide 4
Random Variables Contd
Continuous Random Variables
A continuous random variable can take on any values on a scale.
Examples are distance traveled, time taken to go from one place
to another, heights of individuals, weights of individuals,
temperature of cities, etc.
o------------------------------------------o
A continuous random variable can take on any value on the above
scale
IS 310 – Business Statistics
Slide 5
Random Variables
Question
Family
size
Random Variable x
x = Number of dependents
reported on tax return
Type
Discrete
Distance from x = Distance in miles from
home to store
home to the store site
Continuous
Own dog
or cat
Discrete
x = 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)
IS 310 – Business Statistics
Slide 6
Discrete Probability Distributions
The probability distribution for a random variable
describes how probabilities are distributed over
the values of the random variable.
We can describe a discrete probability distribution
with a table, graph, or equation.
IS 310 – Business Statistics
Slide 7
Discrete Probability Distribution
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Let’s consider the illustration in Section 5.2 (10-Page
190; 11-Page 197)
DiCarlo Motors in Saratoga, New York sold the
following number of cars over the past 300 days:
0 cars on 54 days; 1 car on 117 days; 2 cars on 72 days;
3 cars on 42 days; 4 cars on 12 days; and 5 cars on 3
days.
The probability distribution is shown in Table 5.3 (10Page 191; 11_Page 198).
IS 310 – Business Statistics
Slide 8
Discrete Probability Distribution
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Table 5.3
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Number of cars sold
0
1
2
3
4
5
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IS 310 – Business Statistics
Probability
54/300 = 0.18
117/300 = 0.39
72/300 = 0.24
42/300 = 0.14
12/300 = 0.04
3/300 = 0.01
Slide 9
Sample Problem
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Problem # 8 (10-Page 193; 11-Page 200)
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Number of operating rooms used over a 20-day
period.
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Number of Rooms
1
2
3
4
IS 310 – Business Statistics
Frequency
3
5
8
4
Probability
3/20 = 0.15
5/20 = 0.25
8/20 = 0.40
4/20 = 0.20
Slide 10
Expected Value and Variance
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The expected value of a random variable is obtained by multiplying
each value of the random variable by its probability and adding the
resulting products.
Let’s refer to the problem of car sales of DiCarlo Motors. Look at Table
5.5 (10-Page 196) or Table 5.4 (11-Page 203).
No. of Cars Sold (x)
0
1
2
3
4
5
Probability [f(x)]
0.18
0.39
0.24
0.14
0.04
0.01
x. f(x)
0
0.39
0.48
0.42
0.16
0.05
Expected Value of x = E(x) = 1.50
IS 310 – Business Statistics
Slide 11
Expected Value and Variance
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What does Expected Value mean?
Expected Value is the average value of the random
variable over a long period of time.
Referring to DiCarlo Motors, the Expected Value of
1.5 means that DiCarlo can expect to sell, on the
average, 1.5 cars per day over a long period of time.
IS 310 – Business Statistics
Slide 12
Expected Value and Variance

The variance of a random variable is obtained by
using formula 5.5 (10-Page 196; 11-Page 203).
Calculations are shown in Table 5.6 (10-Page 197) or
Table 5.5 (11-Page 204).

The variance is calculated as 1.25 so the standard
deviation is √ 1.25 = 1.118.
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IS 310 – Business Statistics
Slide 13
Binomial Probability Distribution
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Two discrete probability distributions that we will
study are:
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Binomial Probability Distribution
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Poisson Probability Distribution
IS 310 – Business Statistics
Slide 14
Binomial Distribution
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Four Properties of a Binomial Experiment
1. The experiment consists of a sequence of n
identical trials.
2. 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.
stationarity
assumption
4. The trials are independent.
IS 310 – Business Statistics
Slide 15
Binomial Distribution
Our interest is in the number of successes
occurring in the n trials.
We let x denote the number of successes
occurring in the n trials.
IS 310 – Business Statistics
Slide 16
Binomial Distribution
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Binomial Probability Function
n!
f (x) 
p x (1  p)( n  x )
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
IS 310 – Business Statistics
Slide 17
Binomial Distribution

Binomial Probability Function
n!
f (x) 
p x (1  p)( n  x )
x !(n  x )!
Number of experimental
outcomes providing exactly
x successes in n trials
IS 310 – Business Statistics
Probability of a particular
sequence of trial outcomes
with x successes in n trials
Slide 18
Example of Binomial Distribution
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Martin Clothing Store (10-Page 202; 11-Page 209))
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Given: The probability of a customer making a purchase is 0.3. Three
customers walk into the store.
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What is the probability that two of the three customers will make a
purchase?
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This is an example of binomial distribution for the following reasons:
1. There are only two outcomes: making a purchase (success) or not
making a purchase (failure).
2. The probability of success is 0.3 . There are three trials (three
customers) and we are trying to determine the probability of two
successes.
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IS 310 – Business Statistics
Slide 19
Example of Binomial Distribution
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Martin Clothing Store Problem
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Let’s look at Figure 5.3 (10-Page 203; 11-Page 210).
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Formula 5.8 (10-Page 205; 11-Page 212) can be used to
calculate the probability of two customers making a
purchase.
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n
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P(x=2) = (
x
)p
n-x
(1 – p)
= 0.189
x
IS 310 – Business Statistics
Slide 20
Sample Problem on Binomial Distribution
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Martin Clothing Store Problem
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Rather than using formula 5.8, we could use Table 5
of Appendix B (10- Pages 930-937; 11-Pages 989-997)
to obtain directly the value of any probability
without any calculations.
We need to know the values of p, x, and n to use
Table 5.
For x=2, n=3, and p=0.3, the value of P(x=2) = 0.189
from (10-Page 932; 11-Page 992).
IS 310 – Business Statistics
Slide 21
Sample Problems
Problem # 29 (10-Page 209; 11-Page 216)
Given:
p = 0.30
x = 3 (number of workers who
take public transportation)
n = 10 (total number of workers in the sample)
a. f(3) = 0.2668 (From Table 5 in Appendix B)
b. f(3 or more) = f(3) + f(4) + f(5) + f(6) + f(7) + f(8) + f(9)
+ f(10) = 0.2668 + 0.2001 + 0.1029 + 0.0368 + 0.0090 +
0.0014 + 0.0001 + 0.0000 = 0.62
IS 310 – Business Statistics
Slide 22
Poisson Distribution
A Poisson distributed random variable is often
useful in estimating the number of occurrences
over a specified interval of time or space
It is a discrete random variable that may assume
an infinite sequence of values (x = 0, 1, 2, . . . ).
IS 310 – Business Statistics
Slide 23
Poisson Distribution
Examples of a Poisson distributed random variable:
the number of knotholes in 14 linear feet of
pine board
the number of vehicles arriving at a
toll booth in one hour
IS 310 – Business Statistics
Slide 24
Poisson Distribution
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Two Properties of a Poisson Experiment
1. The probability of an occurrence is the same
for any two intervals of equal length.
2. The occurrence or nonoccurrence in any
interval is independent of the occurrence or
nonoccurrence in any other interval.
IS 310 – Business Statistics
Slide 25
Poisson Distribution

Poisson Probability Function
f ( x) 
 x e 
x!
where:
f(x) = probability of x occurrences in an interval
 = mean number of occurrences in an interval
e = 2.71828
IS 310 – Business Statistics
Slide 26
Poisson Distribution

Rather than using formula 5.11, one could use Table 7
of Appendix B (10-Pages 939-944; 11-Pages 999-1004)
to calculate any probability. We need to know the
values of µ and x to use Table 7.
IS 310 – Business Statistics
Slide 27
Sample Problem
Problem # 40 (10-Page 213; 11-Page 220)
a. Given µ = 48 per hour = 4 per five-minute
f(3) = 0.1954 (From Table 7 in Appendix B)
Given µ = 12 per 15-minute
f(10) = 0.1048 (From Table 7 in Appendix B)
b.
c.
d.
4 calls
f(0) = 0.0183
f(0) = 0.0907 with µ = 2.4
IS 310 – Business Statistics
Slide 28
End of Chapter 5
IS 310 – Business Statistics
Slide 29