Strategic Practice and Homework 8 - Projects at Harvard

This makes sense intuitively since if we observe that V = v, thenX and Y arethe independent r.v.s W and Z, shiftedbytheconstantv.(c) Find the joint PMF of X, Y (as a sum).By (b), a good strategy is to condition on V :P (X = x, Y = y) ===1XP (X = x, Y = y|V = v)P (V = v)v=0min(x,y)Xv=0min(x,y)Xv=0P (X = x|V = v)P (Y = y|V = v)P (V = v)exmin(x,y)X= e 3 x+yv(x v)! e (y v)! e v!y v vv=0v(x v)!(y v)!v! ,for x and y nonnegative integers. Note that we sum only up to min(x, y) sincewe know for sure that V apple X and V apple Y .Miracle check: note that P (X =0,Y =0)=P (V =0,W =0,Z =0)=e 3 .7. Let X be Hypergeometric with parameters b, w, n.(a) Find E X 2by thinking, withoutanycomplicatedcalculations.XIn the story of the Hypergeometric, is the number of pairs of draws such2that both balls are white. Creating an indicator r.v. for each pair, we have✓ ◆ ✓ ◆X n w w 1E =2 2 w + b w + b 1 .(b) Use (a) to get the variance of X, confirming the result from class thatVar(X) = NNn1 npq,where N = w + b, p = w/N, q =1 p.5