I am still working on the gamma distribution and first thank you so much for help without help i wouldn't be able to do the previous questions.

Unfortunately, I got stuck with another question T-T

For Z distributed on R^3 with Z1, Z2, Z3 IID Z ~ exp(1)

if I have Z = Z2 / (Z1+Z3) then what is E(Z) and Var(Z)?

Since I was using the chi-square,

I got Z2 ~ exp(1) => Z2 ~ G(1) => 2Z2 ~ G(1,2) => X^2(2)/2

because Y ~ X^2(m) iff y = 2z, z ~ G(m/2) //G is gamma

In the similar manner I got Z1+Z3 ~ G(2) => X^2(4)/2

Thus, Z2 / (Z1+Z3) = [X^2(2)/2] / [X^2(4)/2] which becomes F-dist with (2,4)

Then HOW can I find the expected value of Z and Var(Z)????? Is there any equation to find F-distribution's E value and Variance???

plz help me!!! T-T