I came across a problem where the marginal distribution of a random variable $Y$, $f(y) = c/y^2$ and $f(x|y) = 1/y$.
Can I simply multiply these two to get $f(x,y)$ the joint distribution of $X$ and $Y$, which in this case will be $c/y^3$. And then integrate it over all $Y$ to find the marginal distribution of $X$.
Best Answer
In effect, yes, $f(x,y)=f(x|y)f(y)$, which is essentially just elementary probability rules.
See the relevant section of Wikipedia's article on Conditional probability distribution.
However, you need to take proper care about the limits of $X|Y=y$ (and $Y$ for that matter), since that will define where $(X,Y)$ lives.
Were ranges given?
Then, yes, once you have the joint you could compute a marginal for $X$, but again, you must take care about the limits. Not dealing properly with them is perhaps one of the biggest sources of errors for beginners.