The Mean of a Discrete Random Variable Lesson 7.2.1

Starter An unfair six-sided die comes up 1 on half of its rolls. The other half of its rolls are evenly spread through the other 5 outcomes. If X is the number that comes up, write PDF for X.

Objectives Evaluate the mean of a discrete random variable from its PDF Use the other names and notations for the mean: –µ x –Expected Value –E(x) Define what is meant by a “fair game” California Standard 5.0 Students know the definition of the mean of a discrete random variable and can determine the mean for a particular discrete random variable.

The “Expected Value” of a discrete Random Variable Suppose you roll a fair die 600 times. How many times would you expect that each face comes up? How many TOTAL SPOTS will you see in the 600 rolls? What is the average number of spots per roll? This is the expected value of X, also called E(x), where X is the number of spots that shows on each roll –Note that we do not “expect” to get this result on any one roll (it’s actually impossible in this case) –The expected value is the average result over the long run This is also called the MEAN of the discrete random variable. –The mean is denoted by the Greek letter µ or µ x

The Formula for µ To find the mean (or expected value) of any discrete random variable, multiply each possible outcome by its probability –µ x = x 1 p 1 + x 2 p 2 + … + x i p i –µ x = Σx i p i Apply the formula to a fair die: What is the mean of X when X is defined as the number of spots that show when the die is rolled? What is the mean of X in the starter?

Example Four coins are tossed and X is the number of heads that show. –Recall the PDF of X (it’s already in your notes) –Find µ x by using the formula µ x = (0)(1/16) + (1)(4/16) + (2)(6/16) + (3)(4/16) + (4)(1/16) = 2

Example A certain lottery is played by paying $1 for a chance at a $500 prize. One of 1000 numbered balls is drawn. If your number matches the ball, you win. –Let X be the amount you are paid (ignore the $1) –Write the PDF of X Calculate E(x) –E(x) = (500)(.001) + (0)(.999) =.50 –So you expect to be paid $.50 on average for every $1 game! X5000 P(X)

Now let’s take the $1 into account A certain lottery is played by paying $1 for a chance at a $500 prize. One of 1000 numbered balls is drawn. If your number matches the ball, you win. –Let X be the net amount you win –Write the PDF of X Calculate E(x) –E(x) = (499)(.001) + (-1)(.999) = -.50 –So you expect to lose $.50 on average for every game! What should E(x) be to make the game fair? –By definition, a game is fair if E(X) = 0 X499 P(X)

The Purpose of Odds You roll a fair die and bet $1 that a 4 comes up. –What is the probability that you win the bet? –What are the odds against winning? –How much should you and your opponent bet? Assuming proper odds are used, let X be your net winnings. Write the PDF and find E(X) E(X) = (5)(1/6) + (-1)(5/6) = 0 X+5 P(X)1/65/6

Objectives Evaluate the mean of a discrete random variable from its PDF Use the other names and notations for the mean: –µ x –Expected Value –E(x) Define what is meant by a “fair game” California Standard 5.0 Students know the definition of the mean of a discrete random variable and can determine the mean for a particular discrete random variable.

Homework Read pages 385 – 394 Do problems 17 – 19, 21, 22, 23