PDF of Joint Distribution of $X$ and $Z$ given Joint Distribution of $X$ and $Y$ ; $Y$ and $Z$.

Question

The Joint distribution of random variables $X$ and $Y$ has PDF $f(x, y) = x + y , 0 < x < 1, 0 < y < 1$.

The Joint distribution of random variables $Y$ and $Z$ has PDF $g(y, z) = 3(y + 0.5 )z^2, 0 < y < 1, 0 < z < 1$

Which of the following could be the PDF of the joint distribution of $X$ and $Z$?

1. $x + 1.5z^2 , 0<x,z<1$

2. $x + 0.5 + 3z^2 , 0<x,z<1$

3. $3 (x + 0.5z^2) , 0<x,z<1$

4. $x+z, 0<x,z<1$

I am only able to rule out option 2 as integral over the given domain is not coming out to be 1 , therefore it's not even a PDF. But, for the other options , I am not able to understand how to proceed.

Answer 1

Hint: Determine the marginal PDF of $X$ from the joint PDF of $X$ and $Y$: $$f_X(x)=\int f(x,y) dy$$ where the integral runs from $y=0$ to $y=1$. Similarly determine the marginal PDF $f_Z(z)$ for $Z$ from the joint PDF $g(y,z)$. These marginal PDFs for $X$ and $Y$ should agree with what you get from whichever option is correct. (And for consistency's sake, you should get the same marginal PDF for $Y$ from the joint PDFs $f(x,y)$ and $g(y,z)$.)