Today Today: Begin Chapter 3 Reading: –Covered from Chapter 2 –Please read Chapter 3 –Suggested Problems: 3.2, 3.9, 3.12, 3.20, 3.23, 3.24, 3R5, 3R9

Example Have 50 men, each 26 years old Have 50 women, each 28 years old What is the average age of the 100 people?

Example Have 500 men, each 26 years old Have 50 women, each 28 years old What is the average age of the 100 people?

Example Recall the game show, Let’s Make a Deal A contestant had won $9,000 in prizes, and was offered to exchange the gifts for whatever lay behind one of three doors Behind one door was a $20,000 prize and behind the others were $5,000 and $2,000 prizes, respectively Should the contestant make the exchange?

Expectation (The Mean) If X is a discrete random variable with probability mass function f(x), the expected value (or mean value) is Provided the sum is absolutely convergent (if there are infinitely many values x 1,x 2,… ) Idea, the mean is the weighted average of the possible values of X

Example Recall the game show, Let’s Make a Deal A contestant had won $9,000 in prizes, and was offered to exchange the gifts for whatever lay behind one of three doors Behind one door was a $20,000 prize and behind the others were $5,000 and $2,000 prizes, respectively Should the contestant make the exchange?

Example (True Story) When Derek was a graduate student in Vancouver, parking was $9.00/day If you parked illegally, the ticket was $10.00 Derek discovered that he got a ticket about 50% of the time Which is the better strategy: –Pay $9.00/day –Park illegally

Properties of Expectation For random variables X and Y, –E(c)=c, where c is a constant –E(cX)=cE(X), where c is a constant –E(X+Y) = E(X) + E(Y) –E(aX+bY+c) = aE(X) + bE(Y)+c, where a,b, and c are constants

Example Two dice are rolled – one red and one green Let X be the outcome of the red die and Y be the outcome of the green die Find E(X+Y)

Conditional Mean Conditional distributions also have means The mean will be conditional on the value of another random variable The conditional mean of Y give X=x is

Example Three digits are picked at random, without replacement, from 1,2, …, 8 Let Y denote the largest digit and X denote the smallest Find the probability function for Y Find E(Y) Find probability function for Y given X=3 Find E(Y|X=3)

Expected Value of a Function After observing a random variable, often interested in some function of the random variable The mean value of a function, g, of a random variable X is:

Example (3.1) Consider a random variable X, with probability function f below: Find E(X 2 ) x01234 f(x)

Example (3.10) Recall the game show, Let’s Make a Deal A contestant had won $9,000 in prizes, and was offered to exchange the gifts for whatever lay behind one of three doors Behind one door was a $20,000 prize and behind the others were $5,000 and $2,000 prizes, respectively Suppose that the contestant owes a murderous loan shark $9000, due the next day Can use a utility function to help express thenotion of expectation in this situation

Expected Values and Joint Distributions Some useful relations: How do we get these?

Expected Values and Joint Distributions When random variables X and Y are independent, How did we get this?