Practice(Problems(Ch.7( ( ( Name:(____________________________________(((1) (In this particular game, a player pays $5 and draws a card from a deck. If he draws the ace of hearts, he gets paid $100. For any other ace, he gets $10, and for any other heart, he gets $5. If he draws anything else, he loses. Should you play? Would you play for a top prize of $200? 2) (Suppose a used car dealer runs autos through a two-stage process to get them ready to sell. The mechanical checkup costs $50 per hour and takes an average of 90 minutes, with a standard deviation of 15 minutes. The appearance prep (wash, polish, etc.) costs $6 per hour and takes an average of 60 minutes, with a standard deviation of 5 minutes. a) What are the mean and standard deviation of the total time spent preparing a car?

(Note that we cannot find the standard deviation if we do not believe that the two phases of the process are independent, an important assumption to check.)

b) What are the mean and standard deviation of the total expense to prepare a car?

c) What are the mean and standard deviation of the difference in costs for the two phases of the operation?

3) (Consider a dice game: no points for rolling a 1, 2, or 3; 5 points for a 4 or 5; 50 points for a 6. Find the expected value and the standard deviation.

a) Imagine doubling the points awarded. What are the new mean and standard deviation?

b) Now imagine just playing the game twice. What are the mean and the standard deviation of your total points?

c) Suppose you and a friend both play the dice game. What are the mean and standard deviation of the difference in your winnings?

4) Let X be a discrete random variable with probabiltiy distribution:

X p(x) 1 .15 2 .40 3 .10 4 .35

a)((Find( )2( =XP .((b)((Find( )2( ≤XP .((c)((Find( )2( <XP .((d)((Find( )42( ≤≤ XP .(((

5) A player rolls two dice. The players receive $1.00 for each dot on the face of the two dice. For example, he will be paid $9.00 if the dice show a 4 and a 5. Let X = the number of dollars won each game. a) Is X a discrete or continuous random variable? b) Show the probability distribution of X.

( c) Find the expected value of X. d) Find the variance of the pay off. 6) Let X be a discrete random variable with probability distribution:

X p(x) 1 .10 2 .45 3 .15 4 .25

Does this assignment define a proper probability distribution? Explain. (