1. Chapter 16
Random Variables
Copyright © 2009 Pearson Education, Inc.

2. Expected Value: Center
A random variable assumes a value based on the outcome of a random event.
We use a capital letter, like X, to denote a random variable.
A particular value of a random variable will be denoted with a lower case letter, in this case x.

3. Expected Value: Center (cont.)
There are two types of random variables:
Discrete random variables can take one of a finite number of distinct outcomes.
Example: Number of credit hours
Continuous random variables can take any numeric value within a range of values.
Example: Cost of books this term

4. Expected Value: Center (cont.)
A probability model for a random variable consists of:
The collection of all possible values of a random variable, and
the probabilities that the values occur.
Of particular interest is the value we expect a random variable to take on, notated μ (for population mean) or E(X) for expected value.

5. Expected Value: Center (cont.)
The expected value of a (discrete) random variable can be found by summing the products of each possible value and the probability that it occurs:
Note: Be sure that every possible outcome is included in the sum and verify that you have a valid probability model to start with.

6. Example

7. Chapter 17
Probability Models
Copyright © 2009 Pearson Education, Inc.

8. Bernoulli Trials
The basis for the probability models we will examine in this chapter is the Bernoulli trial.
We have Bernoulli trials if:
there are two possible outcomes (success and failure).
the probability of success, p, is constant.
the trials are independent.

9. The Geometric Model
A single Bernoulli trial is usually not all that interesting.
A Geometric probability model tells us the probability for a random variable that counts the number of Bernoulli trials until the first success.
Geometric models are completely specified by one parameter, p, the probability of success, and are denoted Geom(p).

10. The Geometric Model (cont.)
Geometric probability model for Bernoulli trials: Geom(p)
p = probability of success
q = 1 – p = probability of failure
X = number of trials until the first success occurs
P(X = x) = (1 – p)x-1p

11. Independence
One of the important requirements for Bernoulli trials is that the trials be independent.
When we don’t have an infinite population, the trials are not independent. But, there is a rule that allows us to pretend we have independent trials:
The 10% condition: Bernoulli trials must be independent. If that assumption is violated, it is still okay to proceed as long as the sample is smaller than 10% of the population.

12. Example
The American Red Cross says that about 11% of the U.S. population has Type B blood. A blood drive is being held in your area.
Ex: What is the probability that the fourth blood donor is the first donor with Type B blood?

13. Example
The American Red Cross says that about 11% of the U.S. population has Type B blood. A blood drive is being held in your area.
Ex #2: What is the probability that the first Type B blood donor is among the first four people in line?

14. Example
Through 2/24/2011 NC State’s free-throw percentage was 69.6%. In the game with GaTech what was the probability that the first missed free-throw by NC State occurs on the 5th attempt?
“Success” = missed free throw
Success p = = .304
p(5) =  .304 = .0713

15. The Binomial Model
A Binomial model tells us the probability for a random variable that counts the number of successes in a fixed number of Bernoulli trials.
Two parameters define the Binomial model: n, the number of trials; and, p, the probability of success. We denote this Binom(n, p).

16. The Binomial Model (cont.)
In n trials, there are
ways to have k successes.
Read nCk as “n choose k,” and is called a combination.
Note: n! = n x (n – 1) x … x 2 x 1, and n! is read as “n factorial.”

17. The Binomial Model (cont.)
Binomial probability model for Bernoulli trials: Binom(n,p)
n = number of trials
p = probability of success
(1 – p) = probability of failure
X = number of successes in n trials

18. What Can Go Wrong? Be sure you have Bernoulli trials.
You need two outcomes per trial, a constant probability of success, and independence.
Remember that the 10% Condition provides a reasonable substitute for independence.
Don’t confuse Geometric and Binomial models.

19. What have we learned? Bernoulli trials show up in lots of places.
Depending on the random variable of interest, we might be dealing with a
Geometric model
Binomial model

20. What have we learned? (cont.)
Geometric model
When we’re interested in the number of Bernoulli trials until the next success.
Binomial model
When we’re interested in the number of successes in a certain number of Bernoulli trials.