the first question, i wonder the intever [0,N] or [0,T].
the second, a) you can use cov(aX,bY) = ab cov(X,Y)
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.
First of all there are two ways of writing an exponential rv.
It can be either
or , the first has mean the other .
Next we have Cov(S-T,2S+T)=2V(S)-Cov(S,T)-V(T), yes that's a negative in front of a variance.
From S=X+2Y and T=X-2Y, we have V(S)=1+4=5 and V(T)=1+4=5
Cov(S,T)=Cov(X+2Y,X-2Y)=V(X)-4V(Y)=1-4=-3.
Use MGFs or change variables, which isn't too hard here.
The jacobian is just a constant so using the densities isn't a bad way to go.
Well, for Cov(S-T,2S+T) i simplified to
Cov(4Y,3X+2Y)=Cov(4Y,3X)+Cov(4Y,2Y)
since X and Y are independent, and standard normal RV
Cov(4Y,3X)=0 and Cov(4Y,2Y)=(4*2)*Var(Y)=(8)*(1)=8
Similarily, for Cov(S,T) i worked out to equal -3...
I'm still a bit unsure on the other questions...