Download Assignment #7

MATH 310 Homework #7
(Due Wednesday, 2 November 2016)
Read: Read Meester, Chapter 2, §4.
Problems:
1. (Meester 2.7.14). A gambling book recommends the following strategy for the game of
roulette. It recommends that a gambler bets 1 euro on red. If red appears, then the gambler
should take his or her profit and quit the game. If the gambler loses his or her bet, he should
make additional 1 euro bets on red on each of the next two spins of the roulette wheel and
then quit. Let X denote the gambler’s winnings when he quits.
(a) Find P (X > 0).
(b) Is this a winning strategy? Explain your answer.
(c) Compute E[X].
2. Consider again the St. Petersburg game: you toss a coin until the first head appears, and if
this occurs on the k th toss, you win $2k .
(a) Suppose that the house has finite resources totaling 2m dollars. So you win that total
for m tosses, but also for any greater number of tosses. Show that the expected value
of the game is now (m + 1) dollars.
(b) By this measure, what is the most one should one pay to play against Donald Trump,
whose net worth was recently estimated by Forbes (27 Sept 2016) to be $3.7 billion?
(c) In class, we considered the utility function u(x) = log(x), and found that the expected
√
utility of your winnings was finite. Now suppose instead that u(x) = x. Find the
expected utility of your winnings.
3. (Meester 2.7.8). Let X be a random variable with probability mass function given by p(−4) =
p(4) = 1/16 and p(2) = p(−2) = 7/16. Let Y = aX + b, for certain a and b.
(a) For what values of a and b does Y take values in {0, 1, 2, 3, 4}?
(b) Compute the probability mass function of Y for these choices of a and b.
4. On problem 6 from HW#6, you considered the random variable N , the minimum number
drawn in Keno.
(a) Use the pmf to write an expression for E[N ]. Do not yet evaluate it.
(b) Find a formula for P (N > n), n = 0, 1, . . . .
(c) Use the tail sum formula to compute E[N ]. (Hint. Use the identity from HW #3!)
5. Chuck-a-luck. This gambling game is often found at carnivals. The player bets $1 on her
choice of one of the numbers 1 through 6. Three dice are then rolled. If her number appears,
she gets back the original $1 bet plus $1 for each die showing her number. Otherwise, she
loses her $1.
(a) The operator points out there are 3 chances out of 6 for your number to come up once,
which is already an even bet—and any additional appearances are actually to your favor.
What is wrong with this logic?
(b) Let W be the player’s net winnings. Find the pmf for W and use it to compute E[W ].
(c) Explain why we can express W in terms of “brand name” random variables by writing
W = X − I, where X ∼ Binom(3, 1/6) and I is the indicator of the event {X = 0}. Use
this representation to find E[W ] more quickly.
6. (Meester 2.7.13). Compute the expection of X when X has a negative binomial distribution
X ∼ Nbinom(r, p). [Hint. Express X as a sum.]
7. We roll a tetrahedral (4-sided) die twice. Let X be the sum of the rolls and let Y be the
higher of the two rolls.
(a) Find the pmf of Y and use it to computer E[Y ].
(b) Are X and Y independent?
(c) Let A be the event that the first roll is higher than the second. Are A and the event
X = 5 independent?
(d) Observe that Y = k if and only if both rolls are less than or equal to k. Explain why it
follows that P (Y ≤ k) = k 2 /16, for k = 1, 2, 3, 4. Use this to rederive the pmf.
8. A “run” of heads in a coin tossing sequence is a string of consecutive heads. For example, if
the outcome from 12 tosses is HHTTHHHHTHTH then there are four runs of heads (a run
may contain only one head).
(a) Find the expected number of runs of heads in 12 tosses of a fair coin.
Hint: use indicators to count the number of starts of runs.
(b) Find the expected number of runs of heads or of tails; the above example has 7.
(c) What is the expected number of appearances of HHH? The example has two!
(d) Perform a simulation to check your answer for the HHH problem. The function findruns.R
illustrated in the script on our web page should be helpful.
One way to simulate the vector of tosses is tosses <- sample(0:1, 12, replace=TRUE);
that is, we are sampling with replacement from {0, 1}, interpreting the 1’s as heads. Alternatively, you can set tosses <- rbinom(12, 1, 1/2), which gives 12 observations
from a binomial distribution with n = 1 and p = 1/2. But it is not possible to use to
use rbinom(1, 12, 1/2), which counts the number of heads in one set of 12 tosses but
does not reveal the actual sequence of tosses. (Experiment with these commands a bit
to see what is happening.)