1. Chris Morgan, MATH G160 csmorgan@purdue.edu February 3, 2012 Lecture 11 Chapter 5.3: Expectation (Mean) and Variance 1

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3. Expected Value 3 Question: How do you determine the “value” of a game? Is it better to play Roulette than the Lottery? We are looking for ways of describing random variables. Definition of an Expected Value –The expected value of a random variable X with PMF is given by: –The expected value is a weighted average of the possible values of X, weighted by the probabilities.

4. Expected Value 4 We may interchangeably use the terms mean, average, expectation, and expected value and the notations E(X) or μ Note: The expected value of a random variable can be understood as the long-run-average value of the random variable in repeated independent trials. If you are playing a game, and X is what you win in the game, then E(X) would be your average win if you would play the game many many many many many many many many many times.

5. Example #1 5 Recall from last lecture: xP(x) 0 1/16 1 4/16 2 6/16 3 4/16 4 1/16

6. Fundamental Expected-Value Formula 6 Instead of E(X) we can also compute the expected value of a function of X. – If X is a discrete random variable with PMF and is any real valued function of X, then:

7. Example #2 7 Recall from last lecture: I can also compute: xP(x) 0 1/16 1 4/16 2 6/16 3 4/16 4 1/16

8. Example #3 8 Or I can compute: or even: xP(x) 0 1/16 1 4/16 2 6/16 3 4/16 4 1/16 So then: E(X+3) = E(X) + 3 E(2X) = 2*E(X) E(X2) ≠ E(X)2

9. Expectation in a Linear Operator 9 Let X be a random variable and a, b be constants. Then: Let X1,…,X n be random variables. Then:

10. Variance 10 The definition of variance of a random variable is a measure of the spread of its distribution. It is the expected squared deviation from the mean: where μ = E[X] If we know the pmf of X then we can calculate the variance as follows: We can simplify the variance equation to this:

11. Variance 11 - Var(X) is always non-negative (Var(X) >= 0) - Sometimes, we’ll abbreviate: σ2 - Var(X) is a measure of the spread of the random variable. If Var(X)=0, then the spread is zero, i.e. all the probability is concentrated in one point (nothing is random anymore). - The variance is not measured in the same units that the random variable is measure in. (This is a disadvantage!)

12. Example #4 12 Recall from last lecture: I can also compute: Then:

13. Example #4 13 Theorem: Variance is not a linear operator! Let X be a random variable and a, b, c be constants. Then:

14. Variance 14 If X1, X2, …, Xn are independent, then:

15. Standard Deviation 15 Standard deviation of a random variable X is: Note: unlike variance, standard deviation is measured in the same units

16. Practice #5 16 Let X be a discrete random variable with PMF: E(X)= Var(X)= E(2X-3)= Var(2x-3)= X0123 p(x)0.40.20.30.1

17. Practice #6 17 Let X be a RV with mean μ=5 and variance σ²=9 Find E((X-1) 2 ). Find the standard deviation of X.

18. Practice #7 18 For a game, you tell a friend that if a 6-sided die rolls a 2, you will pay her $2. If the die rolls a 3, she will pay you $3. Any other numbers (so 1, 4, 5, or 6) you pay her a quarter. Let W be the random variable representing your friend’s winnings.

19. Practice #7 19 What is the pmf of W? What is the expected amount of money your friend will win?

20. Practice #7 20 What is the standard deviation of your friend’s winnings?

21. Practice #7 21 If you and your friend played this game 5 times, what would the overall expected value and standard deviation of your friend’s winnings be?

22. Practice #8 22 If Var(Z) = 4, then find: Var(5) = Var(Z+1) = Var(2Z) = Var(aZ + b) = Var(b-aZ) =

23. Practice #8 23 Given that the Var(Y) = 9 and the E(Y) = 4, can we find E(Y 2 )?