Question 3 Relating Random Variables (10+?+5+16=38 credits] A casino offers a new game. Let X N fx be a random variable on (0,1] with pdl'px. Let Y be a random variable on [1, 00) such that Y = UK . A random number 6 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 gumes 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 1 1 mm = Fm? 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] fm' X, that is, ( ) 1 I] R such that for any B 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 [ px(.7:)da: = 1 and px(:|:) 2 0 a You should also briey mention how you came up 1with 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