The Mean of a Discrete Random Variable Lesson

The Mean of a Discrete Random Variable Lesson 7. 2. 1

Starter 7. 2. 1 • 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: – µx – 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 µx

The Formula for µ • To find the mean (or expected value) of any discrete random variable, multiply each possible outcome by its probability – µx = x 1 p 1 + x 2 p 2 + … + xipi – µx = Σxipi • 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 µx by using the formula • µx = (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) X 500 0 – Write the PDF of X P(X) . 001 . 999 • Calculate E(x) – E(x) = (500)(. 001) + (0)(. 999) =. 50 – So you expect to be paid $. 50 on average for every $1 game!

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 X 499 • Calculate E(x) P(X) . 001 -1. 999 – 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

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) X +5 -1 P(X) 1/6 5/6 • E(X) = (5)(1/6) + (-1)(5/6) = 0

Objectives • Evaluate the mean of a discrete random variable from its PDF • Use the other names and notations for the mean: – µx – 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