1. Today Today: Finish Chapter 3, begin Chapter 4 Reading: –Have done 3.1-3.5 –Please start reading Chapter 4 –Suggested problems: 3.24, 4.2, 4.8, 4.10, 4.33, 4.42, 4R3, 4R5

2. Some Useful Properties of Mean and Variances Expectation of a Sum of Random Variables: Variance of a Sum of Random Variables:

3. Example (3.23) X and Y are random variables with the following joint distribution Find cov(X,Y) Find Var(X+Y) Find Var(Y|X=0)

4. Chapter 4 (Some Discrete Random Variables) Sometimes we are interested in the probability that an event occurs (or does not occur) In this case, we are interested in the probability of a success (or failure) In this case, the the random variable of interest, X, takes on one of two possible outcomes (success and failure) coded 1 and 0 respectively The probability of a success is P(X=1)=p The probability of a failure is P(X=0)=1-p=q

5. Bernouilli Distribution The probability function for a Bernoulli random variable is: Can be written as: xf(x) 1p 0q=1-p

6. Bernouilli Distribution Mean: Variance:

7. Characteristics of a Bernoulli Process Trials are independent The random experiment takes on only 2 values (X=1 or 0) The probability of success remains constant

8. Example A fair coin: An unfair coin:

9. Binomial Distribution More often concerned with number of successes in a specified number of trials

10. Example A gambler plays 10 games of roulette. What is the probability that they break even?

11. Binomial Distribution Probability Function:

12. Binomial Distribution Mean: Variance

13. Example To test a new golf ball, 20 golfers are paired together by ability (I.e., there are 10 pairs) One golfer in each pair plays with the new golf ball and the other with an older variety Each pair plays a round of golf together Let X be the number of pairs in which the player with new ball wins the match

14. Example If the new ball performs as well as the old ball, –What is the distribution of X? –Find P(X<=2) –Find the mean and variance of X

15. Example ESP researchers often use Zener cards to test ESP ability A deck of cards consists of equal numbers of five types of cards showing very different shapes Some people believe that hypnosis helps ESP ability A card is randomly sampled from the deck A hypnotized person concentrates on the card and guesses the shape

16. Example 10 students performed this test Each student had three guesses In total there were 10 correct guesses. Is this evidence in favor of ESP ability?

17. Independent Binomials If X and Y are independent binomial random variables with distributions Bin(n 1,p) and Bin(n 2,p) then X+Y has a Bin(n 1 + n 2,p)

18. Discrete Uniform Distribution Consider a discrete distribution, with a finite number of outcomes, where each outcome has the same chance of occurring Suppose the possible outcomes of a random variable, X, are 1,2,…n, with equal probability. What is the probability function for X

19. Discrete Uniform Distribution Mean: Variance

20. Example Fair die: