I'm revising for my probability exam, and I'm in the process of going over some past tutorials. I'm have severe issues with calculating the limits of the integral used to calculate marginal densities. I.e, given a joint probability density function, f(x,y), and asked to calculate f(x) by integrating f(x,y) with respect to dy, I'm struggling to figure out how to calculate the limits for the integral.
I've had no issues with a (except for the fact that if I integrate with respect to y first, I am left with a y at the end of the full integral) but I am stuck on how the limits were calculated for question 2b.
I would appreciate any help whatsoever (preferrably a simple explanation).
Thanks.
Best Answer
We are given that $-y\lt x \lt y\;$ and $\;0\lt y$. So when finding the limits of integration for $f_X(x)=\int f_{X,Y}(x,y)\;dy$, you need to know what are the possible values of $y$, given a particular value of $x$, that satisfy those inequalities.
Here, that depends on whether $x\gt 0\;$ or $\;x\lt 0$.
If $x\gt 0$, the required limits are obvious: $-y\lt x \lt y\;$ means the possible values for $y$ are $x\lt y\lt\infty$. A quick check shows that all inequalities above are satisfied.
However, if $x\lt 0$, then $x\lt y\lt\infty$ can't be right because that means $y$ can be negative. This time the important part of $-y\lt x \lt y\;$ is $-y\lt x$, which implies $y\gt -x$. Again, this satisfies all the given inequalities.