Transcript Lecture 22 - Modeling Discrete Variables

Modeling Discrete
Variables
Lecture 22-1
Sections 6.4
Wed, Mar 1, 2006
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.
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?
Example of a Discrete Variable

We may list each value and its proportion.
For 0.10 of
 For 0.30 of
 For 0.40 of
 For 0.20 of

the population, X = 0.
the population, X = 1.
the population, X = 2.
the population, X = 3.
Example of a Discrete Variable

Or we may present it as a table.
Value of X
0
1
2
3
Proportion
0.10
0.30
0.40
0.20
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
0
1
2
3
Graphing a Discrete Variable

Or we may present it as a histogram.
P(X = x)
0.40
0.30
0.20
0.10
x
0
1
2
3
Discrete Random
Variables
Lecture 22-2
Section 7.5.1
Wed, Mar 1, 2006
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.
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.
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.
Examples of Random Variables

Roll two dice. Let X be the number of sixes.


Roll two dice. Let X be the total of the two
numbers.


Possible values of X = {0, 1, 2}.
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}.
Discrete Probability Distribution
Functions

Discrete Probability Distribution Function (pdf)
– A function that assigns a probability to each
possible value of a discrete random variable.
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 (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.

(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

We can summarize this in a table.
X
0
1
2
P(X = x)
25/36
10/36
1/36
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
0
1
2
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
0
1
2
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?
Example of a Discrete PDF

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

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

Or we may present it as a histogram.
P(X = x)
0.40
0.30
0.20
0.10
x
0
1
2
3