Let the joint pdf of 2 continuous rv's X,Y be f(x,y) = λμe^(-λx-μy) , for (x,y) > 0, and 0 otherwise.

Question

You can see what I did for parts a and b of the question, I believe I did them correctly (if not please let me know). I don't have a solution for this exercise. I am struggling with part c) which asks to find Var(Y) using the total variance formula. I am struggling to find E[Y^2|X] as I somehow set up my integral wrong. Clearly, my integral diverges so I messed up somewhere along the way. I would appreciate any help! Thanks

Answer 1

Here $X$ and $Y$ are independent exponential random variables: $$f(x,y)=\lambda\mu e^{-\lambda x-\mu y}=\lambda e^{-\lambda x}\cdot\mu e^{-\mu y}=:f_X(x)\cdot f_Y(y). $$ The conditional distribution of $Y$ given $X$ (or $X$ given $Y$) is precisely the exponent distribution of $Y$ (of $X$, resp.)

Besides, the condition distribution $f_{Y\mid X}(y\mid x)$ you found at the above is right, but you plugged a wrong distribution into your last integral. This should be $\mu e^{-\mu y}$ instead of $1/\mu$.