Consider the following simplified scenario based on Who Wants to Be a Millionaire?,

a game show in which the contestant answers multiple-choice questions that have 4

choices per question. The contestant (Fred) has answered 9 questions correctly already,

and is now being shown the 10th question. He has no idea what the right answers are

to the 10th or 11th questions are. He has one "lifeline" available, which he can apply

on any question, and which narrows the number of choices from 4 down to 2. Fred has

the following options available.

(a) Walk away with $16,000.

(b) Apply his lifeline to the 10th question, and then answer it. If he gets it wrong, he

will leave with $1,000. If he gets it right, he moves on to the 11th question. He then

leaves with $32,000 if he gets the 11th question wrong, and $64,000 if he gets the

11th question right.

(c) Same as the previous option, except not using his lifeline on the 10th question, and

instead applying it to the 11th question (if he gets the 10th question right).

Find the expected value of each of these options. Which option has the highest expected

value? Which option has the lowest variance?

10. Consider the St. Petersburg paradox (Example 4.3.13), except that you receive $n rather

than $2n if the game lasts for n rounds. What is the fair value of this game? What if

the payo? is $n2?

11. Martin has just heard about the following exciting gambling strategy: bet $1 that a

fair coin will land Heads. If it does, stop. If it lands Tails, double the bet for the next

toss, now betting $2 on Heads. If it does, stop. Otherwise, double the bet for the next

toss to $4. Continue in this way, doubling the bet each time and then stopping right

after winning a bet. Assume that each individual bet is fair, i.e., has an expected net

winnings of 0. The idea is that

1+2+22 + 23 + + 2n = 2n+1

The unit circle {(x, y) : x2 +y2 = 1} is divided into three arcs by choosing three random

points A, B, C on the circle (independently and uniformly), forming arcs between A and

B, between A and C, and between B and C. Let L be the length of the arc containing

the point (1, 0). What is E(L)? Study this by working through the following steps.

(a) Explain what is wrong with the following argument: "The total length of the arcs is

2?, the circumference of the circle. So by symmetry and linearity, each arc has length

2?/3 on average. Referring to the arc containing (1, 0) is just a way to specify one of

the arcs (it wouldn't matter if (1, 0) were re

20. LabTech is a company that manufactures microscopes and other laboratory instruments. On (I point) occasion, one of the microscopes is defective when it comes off of the production line. The probability that a microscope is defective is 0.17%. Find the probability that the first defective microscope comes after the first 100 microscopes off of the production line. 00.156 30.181 00.742 00.844Let Y1, . .. , Yn ~ Poi(1), for n = 1, 2, ... . Show that Xn = In - n D, N (0, 1), VIn as n - too, where T, = > Y, the sample total based on a sample of size n. That is, show i=1 that the limiting distribution of X, is the standard normal distribution. HINT: Re-write X, as Xn = AnZn, for some RVs An and Zn. Then apply the theorems in Sections 5.1 and 5.2, on pp. 289-300 of textbook, possibly in conjunction with the Weak Law of Large Numbers and the Central Limit Theorem, to show that An + 1 and Zn N (0, 1), as n + +oo.18. Suppose goods X and Y are produced along a production possibilities frontier 4X* + Y' = 500 and they are perfect substitutes such that U = X + Y. The slope of the production possibilities -4X frontier is How much X should be produced? 500 - 4X- a. b. 5 c. 10 d. 20 ANS: b 19. Suppose goods X and Y are produced along a production possibilities frontier 4X* + Y' = 500 and they are perfect substitutes such that U = X + Y. The slope of the production possibilities -4X frontier is How much Y should be produced? 500-4X- a. 5 C. 10 20 ANS: d