Transcript Lecture 23 - Random Variables

Random Variables
Lecture 23
Section 7.5.1
Mon, Oct 25, 2004
Random Variables
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Random variable – A variable whose value is
determined by the outcome of an procedure.
The random variable takes on a new value each
time the procedure is performed.
That is why it is “variable.”
Examples of Random Variables
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Roll two dice. Let X be the number of sixes.
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Possible values of X = {0, 1, 2}.
Select a player on the Baltimore Orioles. Let X
be his batting average.
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Possible values of X are
{x | 0 ≤ x ≤ 1}.
Types of Random Variables
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Discrete Random Variable – A random variable
whose set of possible values is a discrete set.
Continuous Random Variable – A random
variable whose set of possible values is a
continuous set.
In the previous two examples, are they discrete
or continuous?
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.
Example of a Discrete PDF
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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)
Why Use a Random Variable?
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We design the sample space so that it will be
easy to find the probabilities.
This may involve more than just the
characteristic in which we are interested.
In the previous example,
We cared about only the number of sixes,
 But we incorportated the order of the two numbers
into the sample space.
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Why Use a Random Variable?
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In the previous example, would it be wrong to
let the sample space be
S = {0, 1, 2},
representing the possible number of sixes?
Why Use a Random Variable?
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The random variable allows us to set up the
sample space in any way that is convenient.
Then, through the random variable, we can
focus on the characteristic of interest.
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.
What is the pdf of X?
Example of a Discrete PDF
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We may list each value.
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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Or we may present it as a chart.
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
Let’s Do It!
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Let’s do it! 7.20, p. 426 – Sum of Pips.