Note: For the purposes of this assignment, if X is a random variable we let px denote the probability density function [pdf) of X, Fx to denote it's cumulative distribution function, and P to denote probabilities. These can all be related as follows: E P[X S r) = Fx[r) 2/ pX[z)dz 00 6 Po. 3 X <_: b fxs fxm px . often we will simply write as p where it clear what random variable the distribution refers to. you should show your derivations but may use a computer algebra system to assist with integration or differentiation. are not assessing ability integrate f differentiate herel casino offers new game. let x fx be on pdf px. y such that number c is sampled from and player guesses m e if guess was lower than then wins dollars means higher pay out more money guessed too high they go bust have dollar. probability density function py for given by ylgpxli hence otherwise compute expected prot under this answer in terms of simplied possible. suppose chooses uniform over a: g :1 : strategy maximise their find> 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