1. Lecture 34 Section 7.5 Wed, Mar 24, 2004
Random Variables
Lecture 34
Section 7.5
Wed, Mar 24, 2004

2. Random Variables
Random variable – A variable whose value is determined by the outcome of an experiment.
The random variable takes on a new value each time the experiment is performed.
That is why it is “variable.”

3. Examples of Random Variables
Roll two dice. Let X = number of sixes.
Possible values of X = {0, 1, 2}.
Select a player on the Baltimore Orioles. Let X = 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; X = no. of sixes.
Draw the 6 x 6 rectangle showing all 36 possibilities.
From it we may determine that
P(X = 0) = 25/36.
P(X = 1) = 10/36.
P(X = 2) = 1/36.

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.
See the previous example.

8. Why Use a Random Variable?
We were interested in the number of sixes.
So why not let the sample space be the possible number of sixes?
S = {0, 1, 2}
Would that be wrong?

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.

16. The Mean and Standard Deviation
Mean of a Discrete Random Variable – The average of the values that the random variable takes on, in the long run.
Standard Deviation of a Discrete Random Variable – The standard deviation of the values that the random variable takes on, in the long run.

17. The Mean of a Discrete Random Variable
The mean is also called the expected value.
However, that does not mean that it is literally the value that we expect to see.
“Expected value” is simply a synonym for the mean or average.

18. The Mean of a Discrete Random Variable
The mean, or expected value, of X may be denoted by either of two symbols.
µ or E(X)
If the random variable is called Y, then we would write E(Y).
Or we could write µY.

19. Computing the Mean Given the pdf of X, the mean is computed as
µ = x1P(X = x1) + … + xnP(X = xn)
=  xiP(X = xi).
This is a weighted average of X.
Each value is weighted by its likelihood.

20. Example of the Mean
Recall the example where X was the number of children in a household. x
P(X = x)
0.10 1
0.30 2
0.40 3
0.20

21. Example of the Mean Multiply each x by the corresponding probability.
P(X = x)
xP(X = x)
0.10
0.00 1
0.30 2
0.40
0.80 3
0.20
0.60

22. Example of the Mean Add up the column of products to get the mean. x
P(X = x)
xP(X = x)
0.10
0.00 1
0.30 2
0.40
0.80 3
0.20
0.60
1.70 = µ

23. Let’s Do It!
Let’s do it! 7.23, p. 430 – Profits and Weather.

24. Assignment Page 442: Exercises 48 – 52, 54.