STA230/MTH230: Probability
Homework #11: Conditional Distributions
Text Problems are from Jim Pitman, Probability.
- Exercises
| § 6.1: | | 1 |
5 |
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| § 6.2: | | 6 |
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| § 6.3: | | 2 |
4 |
8 |
13 |
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| § 6.4: | | 1 |
3 |
5 |
6 |
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- Another Problem
Granny chooses an amount X 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 $Y. 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?
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If Granny used an exponential distribution with mean $10, what strategy
optimizes the expected value of the amount of money you keep?
Good Luck, and Happy Thanksgiving!