Solved – How to choose the bounds for the integral when calculating a marginal distribution from a joint distribution

Question

Let $f(x,y)=2x$ for $0 < x < y < 1$ and zero otherwise. I would like to find the marginal distribution $f_{X}(x)$, which is equal to $$\int f(x,y)dy.$$
However, should integrate over $[0, y]$ or $[0, 1]$? Why?

Answer 1

You need to integrate over the support of $y$ which in this case is $(x,1)$ by the definition of the bounds on $x$ and $y$ given in the question.

To be explicit, in order to obtain the marginal distribution of $X$ you need to solve the following:

$$f_X(x)=\int f_{X,Y}(x,y)dy=\int_x^1 2x dy$$