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Discrete Random Variable Exercises

1. Twenty letters are randomly selected one at a time with replacement. Let V be the number of vowels selected.

A. Give p(v), the probability mass function of V.

B. What is the probability 4 or fewer vowels are selected?

C. Find the mean and standard deviation of V. 2. A mechanic examined the number of cylinders (engine size) and the proportion of customers with each engine size. The resulting table was obtained.

y 4 6 8

p(y) .5 .3 ?

A. What proportion of customers have 8-cylinder engines? (Assume no other engine sizes exist among this mechanic's customers.)

B. Give the cdf, F(y) for this scenario.

C. Suppose the cost of a tune-up, C(X), depends on the engine size according to the equation C(X) = 20 + 3X + 0.5X2. What is the expected tune-up cost among customers?

3. Suppose freezers sold locally come in 13.5, 15.9, or 19.1 cubic feet of storage space. The sells volume indicates that 20% are the smallest size with 30% being the largest (the remaining portion is the medium size). Suppose the industry says the price paid for a freezer is given as P = 25 X - 8.5 (in dollars). How much should the seller expect will be paid by the next customer? (Find E(P).)

4. Suppose that college basketball players make 67% of their free throws. Randomly select 5 players and observe their last attempt (observe 5 random free throw attempts). Let Y be the number of free throws made. Assume each of the 5 events are independent from one another. A. Give p(y). B. Find E(Y) and V(Y). C. Calculate the probability that there are more free throws made than not made.

5. A new game consists of rolling a fair 6-sided die, then flipping a coin the number of times indicated by the die. For example, if you roll a 3, you would flip the coin 3 times. You win the round if you flip at least two heads. Define the random variable Y = # heads flipped (per round). A. Give p(y), the pmf of Y. B. Calculate E(Y) and V(Y). C. Find the probability that a player wins during one play (a round) of this game.