As the title says, I am wondering how to find the joint distribution of two variables when only given the conditional distribution.
An example problem is, Suppose, $Y$ given $X = x$ follows Exponential($1/x$): $f_{Y \mid X}(y\mid X = x) ~=~ (1/x)e^{-(y/x)}$ iff $y > 0$ and $X$ follows Exponential(1) distribution. Find joint distribution of $Y$ and $X$.
I know the joint distribution of two variables is equal to the conditional distribution multiplied by the marginal distribution of the 'given' variable, but I am not sure how to find the marginals from the information given.
Is the marginal of $X$ equal to the probability density function of an Exponential(1) distribution?
Also, how can I find the variance of $Y$?
Would I just use moment generating functions to solve for the variance of Y?