1) Let X₁ be a random variable with probability function given in the table below:

               x₁               1  2  3

fX₁ (x₁) = P(X₁ = x₁) .1 .6 .3

Let X₂ be a random variable with probability function given in the table below:

                x₂            1 2 3

fX₂ (x₂)= P(X₂ = x₂).8 .1 .1

If X₁ and X₂ are independent, then E(X1X2) is closest to
a) 3.0625
b) 2.6147
c) 4.0000
d) 2.8600

2) Two salespersons work for the same ?rm. On any day, the ?rst salesperson makes a sale to any customer with probability 0.3, independently of other customers. The second salesperson makes a sale to any customer with probability 0.4, independently of other customers and independently of the ?rst salesperson. On a particular day, the ?rst salesperson sees a total of 15 customers and the second salesperson sees a total of 22 customers. The expected value of the total number of sales for the two salespersons on that day is closest to

a) 18.5000
b) 8.4300
c) 12.9500
d) 13.3000

3) Let X and Y be random variables with V ar(X)=1.69, V ar(Y )=9. The correlation coefficient between X and Y is ? =0.5. Then Cov(X, Y )is closest to

a) 0.1282
b) 2.7577
c) 1.9500
d) 7.6050

4) Let X and Y be independent random variables with V ar(X)=25, V ar(Y )=16. If U =2X!Y +12, then V ar(U)is closest to

a) 116
b) 66
c) 46
d) 84

5) A certain fast food company has a game based on the game of Monopoly. When purchasing one of their meals, you are given a sticker with one of the 28 possible properties, utilities or railroads that are part of the game. Assume that there are a very large number of stickers available, that there are equal numbers of each of the 28 possible stickers, and that they are randomly mixed. Then the expected number of meals that you must buy in order to get the complete set of 28 stickers is

a) 406.0000
b) 514.6174
c) 109.9608
d) 68.6000

6) Cars approaching an intersection turn left with probability 3/15, go straight through with probability 7/15, or turn right with probability 5/15. We assume independence from car to car. De?ne random variables X1, X2, X3 to be the number of cars in a sequence of 100 cars that turn left, go straight through, or turn right, respectively. The value of Cov(X1,X2)is closest to

a) 0
b) -9.3333
c) -6.6667
d) -15.5556

7) Two containers each contain three balls. In each container one ball is marked with the number -1, one is marked with the number 0, and one is marked with the number 1. The balls are thoroughly mixed and one ball is drawn at random from each container. Let L be a random variable giving the largest of the two numbers drawn. Let S be a random variable giving the smallest of the two numbers drawn. If the same number is drawn from each container, then L = S.The value of Cov(L, S)is closest to

a) 0.4444
b) -0.1975
c) 0
d) 0.1975