1. Lecture 23 Section 7.5.1 Mon, Oct 25, 2004
Random Variables
Lecture 23
Section 7.5.1
Mon, Oct 25, 2004

2. Random Variables
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.”

3. Examples of Random Variables
Roll two dice. Let X be the number of sixes.
Possible values of X = {0, 1, 2}.
Select a player on the Baltimore Orioles. Let X be his batting average.
Possible values of X are
{x | 0 ≤ x ≤ 1}.

4. Types of Random Variables
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?

5. Discrete Probability Distribution Functions
Discrete Probability Distribution Function (pdf) – A function that assigns a probability to each possible value of a discrete random variable.

6. Example of a Discrete PDF
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
P(X = 0) = 25/36.
P(X = 1) = 10/36.
P(X = 2) = 1/36.
(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)

7. Why Use a Random Variable?
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.

8. Why Use a Random Variable?
In the previous example, would it be wrong to let the sample space be
S = {0, 1, 2},
representing the possible number of sixes?

9. Why Use a Random Variable?
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.

10. Example of a Discrete PDF
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?

11. Example of a Discrete PDF
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

12. Example of a Discrete PDF
Or we may present it as a chart. x
P(X = x)
0.10 1
0.30 2
0.40 3
0.20

13. Example of a Discrete PDF
Or we may present it as a stick graph.
P(X = x)
0.40
0.30
0.20
0.10 x 1 2 3

14. Example of a Discrete PDF
Or we may present it as a histogram.
P(X = x)
0.40
0.30
0.20
0.10 x 1 2 3

15. Let’s Do It!
Let’s do it! 7.20, p. 426 – Sum of Pips.