X and Y are two independent random variables such that X is

X and Y are two independent random variables such that X is exponentially distribuited with parameter 1, and Y is uniformly distributed in the interval [0,1]. Compute E[(X + Y)²].

Solution

E(x+y)² =E(x²)+2E(xy)+E(y^2)

As x is exponentially distributed with parameter 1,

E(X^2) = Var(x)+ Mean^2 = 1+1 =2

E(Y^2) = 1/12+ 1 =13/12

E(XY) = E(X) E(Y) as they are independent

= 1

Thus

E(x+y)² =E(x²)+2E(xy)+E(y^2) = 2+13/12+ 1=