13.) Suppose X has the following probability distribution.

x -1 0 1

P(X=x) .50 .35 .15

a) Verify that this is a valid probability distribution.

b) Find the mean of X.

c) Find the variance of X.

d) Based on your answer from part (b), what is the mean of 7X + 6

e) Suppose another random variable Y has a mean of -2. What is the mean of 2X + Y?

14.) Data on 1,000 executives at a marketing firm were collected and are summarized below. There are 3 levels of salary at this company.

X 30,000 60,000 90,000

P(x) .45 .40 .15

a) What is the mean salary at this company?

b) If everyone is given a 15,000 dollar raise, what is the new expected salary?

c) If everyone takes a pay cut of 5,000 dollars, what is the new expected salary?

d) What is the probability that a worker here makes 90,000 dollars?

e) What is the probability that a worker here makes more than 90,000 dollars? What is the probability that a worker here makes less than 90,000 dollars?

15.) Consider the follow probability distribution. x -2 -1 0 1 2

p(X=x) 0.1 0.2 0.4 0.2 0.1

a) Show that this is a valid probability distribution using the two properties from lecture.

b) Find x

c) Find 100

d) Based on (b) and (c), what can you say about the value aX, where is any number.

e) What is P(X = 0)?

f) What is P(X > 0)?

g) What is P(X 0)?

h) What is P(X -2)?

i) What is P(X > 2)?

16.) Brandt Data Solutions is a retail computer store that sells network cables in retail lengths of 3, 5, 10 and 25 feet. From past records they know that the quantity of sales follows the following probability distribution.

x 3 5 10 25

P(X=x) 0.1 0.2 0.5 0.2

a) Find Xand interpret it.

b) What does P(X 5) mean in the context of this problem? Also find P(X 5).

c) What does P(X > 5) mean in the context of this problem? Also find P(X > 5).

d) What is the chance that a randomly selected cable is 25 feet long?

e) What is the chance that a randomly selected cable is at most 10 feet long?

f) Suppose we choose two cables at random. What is the chance they are both at most 10 feet long?