18. Suppose X1, X2, ..., Xn is a random sample from the uniform distribution

on (0, 1) and Z be the sample range. What is the probability that Z is less

than or equal to 0.5?

19. Let X1, X2, ..., X9 be a random sample from a uniform distribution on

the interval [1, 12]. Find the probability that the next to smallest is greater

than or equal to 4?

20. A machine needs 4 out of its 6 independent components to operate. Let

X1, X2, ..., X6 be the lifetime of the respective components. Suppose each is

exponentially distributed with parameter ?. What is the probability density

function of the machine lifetime?

21. Suppose X1, X2, ..., X2n+1 is a random sample from the uniform distribution on (0, 1). What is the probability density function of the sample

median X(n+1)

. An insurance company writes policies for a large number of newly-licensed

drivers each year. Suppose 40% of these are low-risk drivers, 40% are

moderate risk, and 20% are high risk. The company has no way to know which

group any individual driver falls in when it writes the policies. None of the

low-risk drivers will have an at-fault accident in the next year, but 10% of the

moderate-risk and 20% of the high-risk drivers will have such an accident. If a

driver has an at-fault accident in the next year, what is the probability that he or

she is high-risk?

25. You are to participate in an exam for which you had no chance to study, and for

that reason cannot anything but guess for each question (all questions being

of the multiple choice type, so the chance of guessing the correct answer for

each question is 1/d, d being the number of options (distractors) per question;

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offers you the opportunity to choose amongst the following exam formats: I. 6

questions of the 4-choice type; you pass when 5 or more answers are correct;

II. 5 questions of the 5-choice type; you pass when 4 or more answers are

correct; III. 4 questions of the 10-choice type; you pass when 3 or more

answers are correct. Rank the three exam formats according to their

attractiveness. It should be clear that the format with the highest probability to

pass is the most attractive format. Which would you choose and why?

26. Consider the question of whether the home team wins more than half of its

games in the National Basketball Association. Suppose that you study a simple

random sample of 80 professional basketball games and find that 52 of them

are won by the home team.

a. Assuming that there is no home court advantage and that the home team

therefore wins 50% of its games in the long run, determine the probability that

the home team would win 65% or more of its games in a simple random

sample of 80 games.

b. Does the sample information (that 52 of a random sample of 80 games are

won by the home team) provide strong evidence that the home team wins more

than half of its games in the long run? Explain.

27. A refrigerator contains 6 apples, 5 oranges, 10 bananas, 3 pears, 7 peaches, 11

plums, and 2 mangos.

a. Imagine you stick your hand in this refrigerator and pull out a piece of fruit

at random. What is the probability that you will pull out a pear?

b. Imagine now that you put your hand in the refrigerator and pull out a piece

of fruit. You decide you do not want to eat that fruit so you put it back into the

refrigerator and pull out another piece of fruit. What is the probability that the

first piece of fruit you pull out is a banana and the second piece you pull out is

an apple?

c. What is the probability that you stick your hand in the refrigerator one time

and pull out a mango or an orange

May 15. 2019 5. Use the joint cumulative distribution function to answer the questions below. (24 points) 0 Fx.Y(x,y) = 2