1. Probability Continued Chapter 6
Random Variables
Probability Continued
Chapter 6

2. Random Variables
Suppose that each of three randomly selected customers purchasing a hot tub at a certain store chooses either an electric (E) or a gas (G) model. Assume that these customers make their choices independently of one another and that 40% of all customers select an electric model. The number among the three customers who purchase an electric hot tub is a random variable. What is the probability distribution?

3. Random Variable Example
X = number of people who purchase electric hot tub X
P(X)
.216
.432
.288
.064
GGG
(.6)(.6)(.6)
EEG
GEE
EGE
(.4)(.4)(.6)
(.6)(.4)(.4)
(.4)(.6)(.4)
EGG
GEG
GGE
(.4)(.6)(.6)
(.6)(.4)(.6)
(.6)(.6)(.4)
EEE
(.4)(.4)(.4)

4. Random Variables
A numerical variable whose value depends on the outcome of a chance experiment is called a random variable.
discrete versus continuous

5. Discrete vs. Continuous
The number of desks in a classroom.
The fuel efficiency (mpg) of an automobile.
The distance that a person throws a baseball.
The number of questions asked during a statistics final exam.

6. Discrete versus Continuous Probability Distributions
Discrete Properties:
For every possible x value, 0 < p < 1.
Sum of all possible probabilities add to 1.
Continuous Properties:
Often represented by a graph or function.
Can take on any value in an interval.
Area of domain is 1.

7. Means and Variances
The mean value of a random variable X (written mx ) describes where the probability distribution of X is centered.
We often find the mean is not a possible value of X, so it can also be referred to as the “expected value.”
The standard deviation of a random variable X (written sx )describes variability in the probability distribution.

8. Mean of a Random Variable Example
Below is a distribution for number of visits to a dentist in one year.
X = # of visits to the dentist.
Determine the expected value, variance and standard deviation.

9. Formulas
Mean of a Random Variable
Variance of a Random Variable

10. Mean of a Random Variable Example
0(.1) + 1(.3) + 2(.4) + 3(.15) + 4(.05)
= 1.75 visits to the dentist

11. Variance and Standard Deviation
Var(X) =
(0 – 1.75)2(.1) +
(1 – 1.75)2(.3) +
(2 – 1.75)2(.4) +
(3 – 1.75)2(.15) +
(4 – 1.75)2(.05)
= .9875

12. Developing Transformation Rules
Consider the following distribution for the random variable X:

13. X+1
What is the probability distribution for X+1?

14. 2X
What is the probability distribution for 2X?

15. Consider Suppose that E(X) = 2.5, Var(X) = 0.2
What is E(X+5) = ?, Var(X+5) = ?
E(X+5) = = 7.5
Var(X+5) = 0.2 (no change)
What is E(X – 2.2) = ?, Var(X – 2.2) = ?
E(X – 2.2) = 2.5 – 2.2 = 0.3
Var(X – 2.2) = 0.2 (no change)

16. Consider Suppose that E(X) = 2.5, Var(X) = 0.2
What is E(3X) = ?, Var(3X) = ?
E(3X) = 3*2.5 = 7.5
Var(3X) = 32 * 0.2 = 1.8
What is E(2X – 1) = ?, Var(2X – 1) = ?
E(2X – 1) = 2(2.5) – 1 = 4
Var(2X – 1) = 22 * 0.2 = 0.8

17. Transforming Rules
If X is a random variable and a and b are fixed numbers, then ma + bX = a + bmX
If X is a random variable and a and b are fixed numbers, then s2a + bX =b2s2X