# Finding the marginal density function limits of integration

If $f(x,y) = \frac{12}{5x (2-x-y)}$ for 0 < x and y < 1, compute the marginal density function $f_Y(y)$

I know that I have to integrate the joint density function with respect to x, but how I figure out the limits of integration? Also, if the bounds were 0 < y < x < 1, what would the limits be then?

Thanks :D

• you just gave the limits: $x>0, y>1$ – Alex Feb 12 '14 at 2:06
• You must have meant $f_Y(y)$ rather than $f_y(Y)$. – Michael Hardy Feb 12 '14 at 2:11
• fixed, thanks. But my professor said the limits for the marginal density of y for the first case is from 0 to 1, which I don't understand... – girlrockingguna Feb 12 '14 at 2:19
• Your wording "for $0 < x$ and $y < 1$" is ambiguous in that it can be taken to mean that the expression applies for all $x$ and $y$ such that $x$ is positive and $y < 1$. It is most likely that what is meant, and what might even be what your professor actually wrote, is "$0 < x, y < 1$" which is interpreted as shorthand for the more cumbersome $0 < x < 1$ and $0 < y < 1$. So, the limits of integration would be $y=0$ ands $y=1$. – Dilip Sarwate Feb 12 '14 at 2:27
• @DilipSarwate: he did write it like "0<x,y<1" So that's shorthand then? What if he had written "0 < x, y < 2". Then would the bounds be 0 < x < 2 and 0 < y < 2? – girlrockingguna Feb 12 '14 at 3:10