STA230/MTH230: Probability

Homework #11: Conditional Distributions

Text Problems are from Jim Pitman, Probability.

• Exercises
  

  § 6.1:		1	5
  § 6.2:		6
  § 6.3:		2	4	8	13
  § 6.4:		1	3	5	6

• Another Problem
  Granny chooses an amount X≥0 according to probability density function p(x) and puts $X into one envelope and $2X into another. You choose one envelope at random (with probability 50% each) and open it to find some amount $Y. You don't know whether $Y is $X or $2X. You have the opportunity to keep the amount in the envelope, or to swap and keep whatever is in the other envelope (if you trade, you will keep either $Y/2 or $2Y, depending on whether Y=2X or Y=X).
  • What is the conditional probability that your envelope holds $X (and hence it would be advantageous to trade), conditional on the observed value of $Y?
  • For the naive estimate of 1/2 to be correct for that probability, what would Granny's distribution p(x) have to be? Any problems with that?
  • If Granny used an exponential distribution with mean $10, what strategy optimizes the probability you get the larger amount?
  • If Granny used an exponential distribution with mean $10, what strategy optimizes the expected value of the amount of money you keep?

Good Luck! This is the last HW, due Wed Nov 28