1. Modeling Discrete Variables
Lecture 22-1
Sections 6.4
Wed, Mar 1, 2006

2. Two Types of Variable
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.

3. Example of a Discrete Variable
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?

4. Example of a Discrete Variable
We may list each value and its proportion.
For 0.10 of the population, X = 0.
For 0.30 of the population, X = 1.
For 0.40 of the population, X = 2.
For 0.20 of the population, X = 3.

5. Example of a Discrete Variable
Or we may present it as a table.
Value of X
Proportion
0.10 1
0.30 2
0.40 3
0.20

6. Graphing a Discrete Variable
Or we may present it as a stick graph.
P(X = x)
0.40
0.30
0.20
0.10 x 1 2 3

7. Graphing a Discrete Variable
Or we may present it as a histogram.
P(X = x)
0.40
0.30
0.20
0.10 x 1 2 3

8. Discrete Random Variables
Lecture 22-2
Section 7.5.1
Wed, Mar 1, 2006

9. Random Variables
Random variable – A variable whose value is determined by the outcome of a procedure.
The procedure includes at least one step whose outcome is left to chance.
Therefore, the random variable takes on a new value each time the procedure is performed, even though the procedure is exactly the same.

10. 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.

11. A Note About Probability
The probability that something happens is the proportion of the time that it does happen out of all the times it was given an opportunity to happen.
Therefore, “probability” and “proportion” are synonymous in the context of what we are doing.

12. Examples of Random Variables
Roll two dice. Let X be the number of sixes.
Possible values of X = {0, 1, 2}.
Roll two dice. Let X be the total of the two numbers.
Possible values of X = {2, 3, 4, …, 12}.
Select a person at random and give him up to one hour to perform a simple task. Let X be the time it takes him to perform the task.
Possible values of X are {x | 0 ≤ x ≤ 1}.

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

14. Rolling Two Dice 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)

15. Rolling Two Dice We can summarize this in a table. X P(X = x) 25/36 1
25/36 1
10/36 2
1/36

16. Example of a Discrete PDF
Or we may present it as a stick graph.
P(X = x)
30/36
25/36
20/36
15/36
10/36
5/36 x 1 2

17. Example of a Discrete PDF
Or we may present it as a histogram.
P(X = x)
30/36
25/36
20/36
15/36
10/36
5/36 x 1 2

18. 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.
Then X is a random variable.
Which step in the procedure is left to chance?
What is the pdf of X?

19. Example of a Discrete PDF
We may present the pdf as a table. x
P(X = x)
0.10 1
0.30 2
0.40 3
0.20

20. 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

21. 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