Chapter 5
Discrete Probability Distributions
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Random Variables
Discrete Probability Distributions
Expected Value and Variance
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.30
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0
1
2
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4
Random Variables
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A random variable is a numerical description of the
outcome of an experiment.
A random variable can be classified as being either
discrete or continuous depending on the numerical
values it assumes.
A discrete random variable may assume either a
finite number of values or an infinite sequence of
values.
A continuous random variable may assume any
numerical value in an interval or collection of
intervals.
Example: JSL Appliances
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Discrete random variable with a finite number of
values
Let x = number of TV sets sold at the store in one day
where x can take on 5 values (0, 1, 2, 3, 4)
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Discrete random variable with an infinite sequence of
values
Let x = number of customers arriving in one day
where x can take on the values 0, 1, 2, . . .
We can count the customers arriving, but there is no
finite upper limit on the number that might arrive.
Random Variables
Question
Family
size
Random Variable x
x = Number of dependents in
family 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)
Discrete Probability Distributions
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The probability distribution for a random variable
describes how probabilities are distributed over the
values of the random variable.
The probability distribution is defined by a
probability function, denoted by f(x), which provides
the probability for each value of the random variable.
The required conditions for a discrete probability
function are:
f(x) > 0
f(x) = 1
We can describe a discrete probability distribution
with a table, graph, or equation.
Example: JSL Appliances
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Using past data on TV sales (below left), a tabular
representation of the probability distribution for TV
sales (below right) 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)
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.25
.20
.05
.10
1.00
Example: JSL Appliances
Graphical Representation of the Probability
Distribution
.50
Probability
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4
Values of Random Variable x (TV sales)
Discrete Uniform Probability Distribution
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The discrete uniform probability distribution is the
simplest example of a discrete probability
distribution given by a formula.
The discrete uniform probability function is
f(x) = 1/N
where:
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N = the number of values the random
variable may assume
Note that the values of the random variable are
equally likely.
Expected Value and Variance
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The expected value, or mean, of a random variable is
a measure of its central location.
E(x) =  = xf(x)
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The variance summarizes the variability in the values
of a random variable.
Var(x) =  2 = (x - )2f(x)
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The standard deviation, , is defined as the positive
square root of the variance.
Example: JSL Appliances
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Expected Value of a Discrete Random Variable
x
0
1
2
3
4
f(x)
xf(x)
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.00
.25
.25
.20
.40
.05
.15
.10
.40
E(x) = 1.20
The expected number of TV sets sold in a day is 1.2
Example: JSL Appliances
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Variance and Standard Deviation
of a Discrete Random Variable
x
x-
(x - )2
f(x)
0
1
2
3
4
-1.2
-0.2
0.8
1.8
2.8
1.44
0.04
0.64
3.24
7.84
.40
.25
.20
.05
.10
(x - )2f(x)
.576
.010
.128
.162
.784
1.660 =  
The variance of daily sales is 1.66 TV sets squared.
The standard deviation of sales is 1.2884 TV sets.
Example: XYZ Electronics
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The supervisor of XYZ would like to know what the average
number of absentees is daily and also what the standard
deviation is of the daily employee absentee rate. Historical data
provides the follow probability distribution:
Daily
Absentees
0
1
2
3
4
5
6
f(x)
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.12
.10
.02
.02
.01
1.00
Example: XYZ Electronics
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Expected Value of a Discrete Random Variable
x
0
1
2
3
4
5
6
f(x)
xf(x)
.50
.00
.23
.23
.12
.24
.10
.30
.02
.08
.02
.10
.01
.06
E(x) = 1.01
The expected number of absentees is 1.01
Example: XYZ Electronics
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Variance and Standard Deviation
of a Discrete Random Variable
x
x-
(x - )2
f(x)
0
1
2
3
4
5
6
-1.01
-0.01
0.99
1.99
2.99
3.99
4.99
1.0201
0.0001
0.9801
3.9601
8.9401
15.9201
24.9001
.50
.23
.12
.10
.02
.02
.01
(x - )2f(x)
.5101
.0000
.1176
.3960
.1788
.3184
.2490
1.7699 =  
The variance of absentees is 1.7699 squared.
The standard deviation of absentees is 1.3304.