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

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.

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

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.

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.

Scratch Off Ticket $0 $1 $5 $10 $20 $50 $1000 P(payout) .7 .15 .1 .03 .01 .009 .001 What is the expected value? Payout $0 $1 $5 $10 $20 $50 $100 P(payout) .55 .2 .12 .07 .04 .01 What is the probability of winning $50 What is the expected value?

Example

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

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.

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.

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).

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.”

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