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

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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.

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.

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

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:

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

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

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:

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:

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!)

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

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

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

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

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)

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.

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.

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

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

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?

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

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