I have a couple of exercise questions i'm having trouble with which any help would be much appreciated.

1. Suppose N is exponentially distributed with parameter lambda. Given N, the R.V X has uniform distribution in the interval [0,N]. Evaluate E(X) and Var(X).

I tried using E(X)=E[E(X|N)] and worked from their but confused myself...

2.Suppose X and Y are independent standard normal variables. Let S=X+2Y and T=X-2Y.

a)Evaluate Cov(S,T) and Cov(S-T,2S+T)

b)Show that (S,T) has bivariate normal distribution

For a), I worked out:

Cov(S,T)=Cov(X+2Y,X-2Y)=Cov(X,X)+Cov(X,-2Y)+Cov(2Y,X)+Cov(2Y,-2Y)

=Var(X)+0+0+2^2(Var(Y))=1+4=5

Cov(S-T,2S+T)=Cov(4Y,3X+2Y)=Cov(4Y,3X)+Cov(4Y,2Y)

=Cov(4Y,2Y)=I dont know

b)I have no idea...

Thanks in advance.