The Mean of a Discrete Random Variable Lecture 23 Section Wed, Oct 18, 2006

The Mean of a Discrete Random Variable Mean of a Discrete Random Variable – The average of the values that the random variable takes on, in the long run. Mean of a Discrete Random Variable – The average of the values that the random variable takes on, in the long run.

The Mean of a Discrete Random Variable The mean is also called the expected value. The mean is also called the expected value. If X is the number of sixes in a roll of two dice, then the mean of X is the expected number of sixes. If X is the number of sixes in a roll of two dice, then the mean of X is the expected number of sixes. However, that does not mean that it is the value that we literally expect to see. However, that does not mean that it is the value that we literally expect to see. “Expected value” is simply a synonym for the mean or average. “Expected value” is simply a synonym for the mean or average.

The Mean of a Discrete Random Variable The mean, or expected value, of X may be denoted by either of two symbols. The mean, or expected value, of X may be denoted by either of two symbols. µ X or E(X) Usually there are no other variables to be confused with X, so we may write µ rather than µ X. Usually there are no other variables to be confused with X, so we may write µ rather than µ X.

Example of the Mean How would we find the mean of the following pdf? How would we find the mean of the following pdf? Why would it be wrong simply to average the numbers 1, 2, 3, and 4? Why would it be wrong simply to average the numbers 1, 2, 3, and 4? xP(x)P(x)

Computing the Mean Given the pdf of X, the mean is computed as Given the pdf of X, the mean is computed as This is a weighted average of X. This is a weighted average of X. Each value is weighted by its likelihood. Each value is weighted by its likelihood.

Example of the Mean Recall the example where X was the number of children in a household. Recall the example where X was the number of children in a household. xP(x)P(x)

Example of the Mean xP(x)P(x) xP(x)xP(x) Multiply each x by the corresponding probability. Multiply each x by the corresponding probability.

Example of the Mean xP(x)P(x) xP(x)xP(x) = µ Add up the column of products to get the mean. Add up the column of products to get the mean.

Example: Powerball Use the handout to calculate the expected value of a Powerball ticket. Use the handout to calculate the expected value of a Powerball ticket