If X is a Uniform 0 b RV and Y is a Normal mu 1 RV under the

If X is a Uniform (0, b) RV and Y is a Normal (mu, 1) RV, under the assumption that X and Y are independent. Find the joint density f(x, y) Find E[aX + bY] Find Var[aX + bY] Find E[X^2 + Y^2]

Solution

2) a) as X and Y are independent,

joint density = [ e^( -0.5*( y - )^2) ] / sqrt( 2* pi) ) * ( 1 / b) for 0 < x < b and y is any number!

b) E ( aX + bY) = a*E(X) + b*E(Y) = a*(b/2) + b* = b( (a/2) + )..

c) var ( aX +bY) = a^2* var(X) + b^2* var(Y) [ Cov term = 0 as they are independent ]

= a^2 * ( b^2 / 12 ) + b^2 * 1 = b^2( (a^2 / 12) + 1 )....

c) E [ X^2 + Y^2 ] = E(X^2) + E(Y^2) = [ var ( X) + ( E(X) ) ^2 ] +   [ var ( Y) + ( E(Y) ) ^2 ]
= (b^2 / 12 ) + ( b^2 /4 ) + 1 + ^2 ..........