Homework 1 due Thurs Sep 8

1) Imagine in our "envelope game" (in which the player sees one envelope and pays twice that for the other, where the large envelope has 10 times the value of the small) that the algorithm for stuffing the smaller envelope has probability density pd(x|small) = 1 for x between 0 and 1, and pd(x|small) = 0 for x above 1 (so the small envelope has equal chance of any value between 0 and 1, and is not above 1). Note also that this is properly normalized-- if you integrate pd(x|small) over all x from 0 to infinity, you get 1 as required for a probability density.
a) Find pd(x|large), the probability density that if the envelope the player initially looks at is the large envelope, the player will see x. Make sure this is also properly normalized.
b) Find Pd(x), the probability distribution that players who play this game will see x when they choose an envelope to look at. (Recall that Pd(x) = pd(x|small)*Pm(small) + pd(x|large)*Pm(large) where Pm(small) and Pm(large) are the chances the player has selected to look at the small or the large envelope, with no condition on what x they saw because Pm(small) and Pm(large) are not functions of x.)
c) Use the Bayes theorem to find pm(small|x), the probability density over x that if x is seen, it is the small envelope that was selected to look at. Use your result to test the initial expectation of the class that seeing value x cannot change the 1/2 chance this is the small envelope if the player has no idea how much is in the envelopes.
d) Use pm(small|x) to find the expectation value of how much the player will win (which is negative if they lose), given that they have seen x. (The expectation value conditioned on x is
(x) = pm(small|x)*(-2x+10x) + pm(large|x)*(-2x+x/10) where pm(large|x) = 1 - pm(small|x) because given that x is seen, it must either be the small or large envelope.)
e) Now find the overall expectation value <w> per play by averaging <w>(x) over all x by saying <w> = integral over all x of <w>(x)*Pd(x). Is this a game you want to play?

2) Imagine a general version of our "envelope game" in which the larger envelope has A times the amount in the smaller, and the player must pay B times what they see to buy the other envelope.
a) Find the expectation value <w> of what the player wins every time they play, in terms of the expected value in the small envelope, <y>_small, and the expected value of the large envelope, <y>_large. (Do this using the fact that the player has a 1/2 chance of seeing either, given that you are not using the x they saw in your expression here.)
b) Simplify (a) to be just in terms of <y>_small by using the simple relation between <y>_large and <y>_small involving A.
c) Use your results above to find the value of B that makes the game "fair" to the player (i.e., makes the expected winning <w>=0). Did you need to know the value of <y>_small to know the fair B?
d) Interpret why your answer to (c) makes sense, taking the perspective of a player who only plays the game once and has no prior information about how much is in the envelopes.

3) Two major ramifications of the shift from the geocentric to the heliocentric model of the solar system are that it made the other planets to be somewhat like the Earth, and it made the stars in the night sky to be somewhat like the big one in the day sky.
a) Take each of these ramifications and describe one way it made it much easier for us to understand the cosmos.
b) Describe from the general Bayesian perspective why it took two millennia for these shifts to occur. Include in your answer the smallness of stellar parallax, retrograde motion, and models of gravity.

4) Use the virial theorem of celestial mechanics to derive Kepler's third law (which relates the time of an orbit to the size of the ellipse, both scaled to the corresponding quantities for the Earth's orbit). You can simplify your answer by considering only the case of circular orbits, but recognize that your argument is general because you could equally well talk about classes of ellipses with the same shape (same eccentricity), when all the lengths are scaled by one factor, and all the times are necessarily scaled by a related factor. (This is called a "scaling analysis," and is very similar to tracking the units of the quantities involved.) Also, say why this means Jupiter's moons must also follow the same expression, merely scaled to the orbit around Jupiter of one of them, perhaps Io.