A casino offers a new game. Let X ~ fX be a random variable on [0,1] with pdf pX. Let Y be a random variable on [1, 00) such that Y = l / X . A random number c is sampled from Y, and the player guesses a number m E [1, 00). If the player's guess m was lower than c, then the player wins m 1 dollars from the casino (which means higher guesses pay out more money). But if the player guessed too high, (m 2 c), they go bust, and have to pay the casino 1 dollar. a) Show that the probability density function py for Y is given by 39141:) = ylgpxli) b) Hence, or otherwise, compute the expected prot for the player under this game. Your answer will be in terms of m and pX, and should be as simplied as possible. c) Suppose the casino chooses a uniform distribution over (0, 1] for X, that is, ( ) 1 0 R such that for any B > 0, there exists a corresponding player guess m such that the expected prot for the player is at least B. (That is, prove that the expected prot for pX, as a function of m, is unbounded.) Make sure that your choice for pX is a valid pdf, i.e. it should satisfy 1 / mods: = 1 and pm) 2 0 D You should also briefly mention how you came up with your choice for pX. Hint: We want X to be extremely biased towards small values, so that Y is likely to be large, and the player can choose higher values of m without going bust