3 Let X Unif 01 and y N 01 What is E 2X y 1 Now if further a

3. Let X Unif (0,1) and y N (0,1). What is E 2X ?y +1)? Now if further assume X and y are independent, what is Var (2X ?y +1)?

Solution

Given that X is Uniform (0,1) and Y ~N(0,1)

Hence E(X) = 1/2 and Var(x) = 1/12

E(Y) = 0 and Var(x) = 1

x and y are independent.

This implies cov (x,y) =0

Hence Var(2x-y+1) = 4 var(x)+1var(y) +var(1)+cov(x,y)

= 4(1/12) +1(1) +0+0

= 4