Joint pdf $\displaystyle f(x,y)=\dfrac{1}{2x^2y}$ for $1\leq x <\infty , \dfrac{1}{x}<y<x$
To find marginal pdf of $X$ and $Y$
Now the solution is given as : $\displaystyle f_x(X)=\int\limits_{1/x}^{x}\dfrac{1}{2x^2y}\,dy$
$\displaystyle f_y(Y)=\int\limits_{1/y}^{\infty}\dfrac{1}{2x^2y}\,dx$
I need help to visualise this space on the $x$-$y$ plane.
I understood the limits of the first marginal pdf : $\dfrac{1}{x}<y<x$
The second marginal pdf seems tricky: The logic given is that the second set of limits can be manipulated from $\dfrac{1}{x}<y<x$ to $x>\dfrac{1}{y}>\dfrac{1}{x}$.
However, it is given that $1\leq x$, so the integral limits starting from $\dfrac{1}{y}$ can end being less than 1 depending on the value of $y$.
The given limits should form an arc shaped area in the $x$-$y$ plane, but the integral seems to take a parabola shaped area.
Please advise.
Thanks.
Support for the joint distribution of $X,Y$ is the shaded region below.
I think the given solution for $f_Y(y)$ is wrong. The picture makes it clear that $0\lt Y$. So the marginal pdf should be split into two cases:
If $0\lt y\leq 1$:
\begin{eqnarray} f_Y(y) = \int_{x=1/y}^{\infty} f_{X,Y}(x,y)\;dx \end{eqnarray}
If $1\lt y$:
\begin{eqnarray} f_Y(y) = \int_{x=y}^{\infty} f_{X,Y}(x,y)\;dx \end{eqnarray}