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Modeling Discrete
Variables
Lecture 22, Part 1
Sections 6.4
Fri, Oct 13, 2006
Two Types of Variable
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Discrete variable – A variable whose set of
possible values is a set of isolated points on the
number line.
Continuous variable – A variable whose set of
possible values is a continuous interval of real
numbers.
Example of a Discrete Variable
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Suppose that 10% of all households have no
children, 30% have one child, 40% have two
children, and 20% have three children.
Select a household at random and let X =
number of children.
What is the distribution of X?
Example of a Discrete Variable
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Method 1: A list.
We may list each value and its proportion.
X = 0 for 0.10 of
 X = 1 for 0.30 of
 X = 2 for 0.40 of
 X = 3 for 0.20 of
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the population.
the population.
the population.
the population.
Example of a Discrete Variable
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Method 2: A table.
We may present the information as a table.
Value of X
0
1
2
3
Proportion
0.10
0.30
0.40
0.20
Graphing a Discrete Variable
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Method 3: A stick graph.
We may present the information as a stick graph.
Proportion
0.40
0.30
0.20
0.10
x
0
1
2
3
Graphing a Discrete Variable
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Method 4: A histogram.
We may present the information as a histogram.
Proportion
0.40
0.30
0.20
0.10
x
0
1
2
3
Discrete Random
Variables
Lecture 22, Part 2
Section 7.5.1
Fri, Oct 13, 2006
Random Variables
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Random variable – A variable whose value is
determined by the outcome of a procedure
where the outcome of at least one step in the
procedure is left to chance.
Therefore, the random variable may take on a
new value each time the procedure is performed,
even though the procedure is exactly the same.
Random Variables
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A random variable is really the same thing as the
variables we studied in Chapter 2 (page 85).
A variable is a quantitative or qualitative
characteristic that can be observed or measured
for each member of a population.
So what makes it random?
Examples of Random Variables
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Select one person at random from a group of 10
men and 20 women.
Let X be the sex of the person selected.
What are the possible values of X?
What are the probabilities of those values?
Which step of the procedure is left to chance?
Examples of Random Variables
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Roll two dice.
Let X be the number of sixes that turn up.
What is the characteristic that is being observed?
What are the possible values of X?
What are the probabilities of those values?
Examples of Random Variables
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Roll two dice.
Let X be the total of the two numbers. What is
the characteristic that is being observed?
What are the possible values of X?
What are their probabilities?
Examples of Random Variables
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A bus arrives at a bus stop every 15 minutes.
You show up at a random time.
Let X be the time you wait until the bus arrives.
What are the possible values of X?
What are their probabilities?
A Note About Probability
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The probability that something happens is the
proportion of the time that it does happen out
of all the times that it was given an opportunity
to happen.
Therefore, “probability” and “proportion” are
synonymous in the context of what we are
doing.
Discrete Probability Distribution
Functions
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Discrete Probability Distribution Function (pdf)
– A function that assigns a probability to each
possible value of a discrete random variable.
Rolling Two Dice
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Roll two dice and let X be the number of sixes.
Draw the 6  6 rectangle showing all 36
possibilities.
From it we see that (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
P(X = 0) = 25/36.
 P(X = 1) = 10/36.
 P(X = 2) = 1/36.
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(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
(3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
(4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
(5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
(6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
Rolling Two Dice
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We can summarize this as a list (Method 1):
P(X = 0) = 25/36.
 P(X = 1) = 10/36.
 P(X = 2) = 1/36.
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Rolling Two Dice
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We can summarize this in a table (Method 2):
X
0
1
2
P(X = x)
25/36
10/36
1/36
Example of a Discrete PDF
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We may present it as a stick graph (Method 3):
P(X = x)
30/36
25/36
20/36
15/36
10/36
5/36
x
0
1
2
Example of a Discrete PDF
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We may present it as a histogram (Method 4):
P(X = x)
30/36
25/36
20/36
15/36
10/36
5/36
x
0
1
2
Rolling Two Dice
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Roll two dice and let X be the sum of the two
numbers.
From it we see that
P(X = 2) = 1/36.
 P(X = 3) = 2/36.
 P(X = 4) = 3/36.
 P(X = 5) = 4/36.
 P(X = 6) = 5/36.
 etc.
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(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
(3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
(4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
(5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
(6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
Example of a Discrete PDF
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Suppose that 10% of all households have no
children, 30% have one child, 40% have two
children, and 20% have three children.
Select a household at random and let X =
number of children.
Then X is a random variable.
What is the pdf of X?
Example of a Discrete PDF
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We may present the pdf as a list.
P(X = 0) = 0.10.
 P(X = 1) = 0.30.
 P(X = 2) = 0.40.
 P(X = 3) = 0.20.
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Example of a Discrete PDF
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We may present the pdf as a table.
x
0
1
2
3
P(X = x)
0.10
0.30
0.40
0.20
Example of a Discrete PDF
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Or we may present it as a stick graph.
P(X = x)
0.40
0.30
0.20
0.10
x
0
1
2
3
Example of a Discrete PDF
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Or we may present it as a histogram.
P(X = x)
0.40
0.30
0.20
0.10
x
0
1
2
3