Just need problem 8.
For the next two problems, consider the metrics D and D' on the space of (nice random variables). Recall that:
DX,Y = E[X - Y]
DX,Y = E[|X - Y|]
Problem 7: Consider a Polya urn that starts out with one blue ball and one red ball. For each i > 1, let X be 1 if the ith draw is blue, and 0 if the ith draw is red. What is DX,X? What is D'X,X?
Problem 8: Suppose two random variables, X and Y, both have expectation 1. X is 0 or 2, each with probability , and Y is 0 with probability 0.999 and 100 with probability 0.001.
a) Suppose you pay $1 to play, and you win X dollars or Y dollars, depending on which game you pick. Which game would a gambling addict be more likely to pick? Which game would a risk-averse person be more likely to pick?
b) Suppose you pick a game and play it 500 times so you pay $500, and then receive an amount of money that depends on the outcomes. If you picked the X game, is it reasonable to think your net profit or loss will be relatively small (say, between $10 and $50)? (Go with your gut here.) What if you picked the Y game?
c) Based on your answers to the previous two parts, which random variable X or Y feels more risky?
Let W be a random variable that is 1 with probability 1. Note that X, W, and Y all have the same expectation.
d) What is DX,W?
e) What is DY,W?
f) What is D'X,W?
g) What is D'Y,W?
Let Z be a random variable that is 0 with probability 0.999999 and one million with probability 0.000001. Note that its expectation is still 1.
h) What is D(Z,W)?
i) What is DZ,1)?
j) What did you notice about how the distances from 1 changed as the random variables changed?
