This is an exam! You can use any written material and the internet, but you can not discuss this exam with ANYONE. The solution has to be done with Latex and uploaded to our shared Dropbox folder by 8pm today.

Problem 1

Say \(X\sim U\{-3,...,3\}\).

  1. Find the kurtosis of X.

  2. Find \(var(|X|)\)

Problem 2

Say \(X\sim Pois(\lambda)\). Find the density, mean and variance of \(Y=X|X>0\).

Problem 3

Say \(X\sim Beta(n+1,1)\). Find a transformation g such that \(Y=g(X)\sim Exp(1)\).

Problem 4

Say \(X\sim Geom(p)\) and \(Y|X=k\sim U[-k,k]\).

  1. Find the mean and variance of Y.

  2. Find the correlation of X and Y.

Problem 5

In some game a player draws three cards at random from a full deck of 52 cards, and collects as many dollars as the number of aces among the three. Assume he can play the game as often as he wants, and let \(S_n\) be his combined winnings after \(n\) rounds. Compute the moment generating function of \(S_n\) and use it to find the smallest n such \(E[S_n]>1\).