Description

1 Variance Proofs
(a) Let X be a random variable. Prove that:
Var(X) ≥ 0
(b) Let X1,…,Xn be random variables. Prove that:
n
Var(X1 +···+Xn) = ∑ Var(Xi)+2 ∑ cov(Xi,Xj)
i=1 1≤i<j≤n
Hint: Without loss of generality we can assume that E[X1] = ··· = E[Xn] = 0. Why?
(c) Let a1,…,an ∈ R, and X1,…,Xn be random variables. Prove that:
n
ai ·aj ·cov(Xi,Xj) ≥ 0
i=1 1≤i<j≤n
2 Subset Card Game
Jonathan and Yiming are playing a card game. Jonathan has k > 2 cards, and each card has a real number written on it. Jonathan tells Yiming (truthfully), that the sum of the card values is 0, and that the sum of squares of the values on the cards is 1. Specifically, if the card values are c1,c2,…,ck, then we have 0 and
The cards are then going to be dealt randomly in the following fashion: for each card in the deck, a fair coin is flipped. If the coin lands heads, then the card goes to Yiming, and if the coin lands tails, the card goes to Jonathan. Note that it is possible for either player to end up with no cards/all the cards.
Calculate Var(S), where S is the sum of value of cards in Yiming’s hand. The answer should not include a summation.
3 Variance
A building has n upper floors numbered 1,2,…,n, plus a ground floor G. At the ground floor, m people get on the elevator together, and each person gets off at one of the n upper floors uniformly at random and independently of everyone else. What is the variance of the number of floors the elevator does not stop at?