Marginal density function understanding Given a plane with three points, $(0, -1)$, $(2,0)$, and $(0, 1)$ with $x$-axis and $y$-axis connecting three points to make a triangle. Suppose this triangle represents the support for a joint continuous probability density 
$$f(y) = \int f(x,y) \, dx.$$
What can we say about the marginal density $f(y)$? Are they going to be increasing function, decreasing function, neither increasing nor decreasing function, or cannot be identified because of lack of information. 
 A: It may not be enough. One may be expected to give concrete examples. 
By making the joint density $0$ on suitable horizontal strips, say $y=0$ to $y=0.5$, also $y=1.2$ to $y=1.4$, $y=3$ to $y=4$, and constant elsewhere, you can make the marginal density do weird jumping up and down. The simplest example would have say joint density $0$ for $-1\lt y\lt 1$ and an easily evaluated constant on the rest of the triangle. Then you can make an explicit computation of the marginal density of $Y$. 
If one wants the joint density to be continuous (a very unlikely requirement), then some modification would be necessary. 
A: My new understanding is that
If we have a joint (continuous) probability density function = 0 such that $F(y)=\begin{cases} \int_0^{2y+10} 0 dx,& -5\le x \le 0\\ \int_0^{10-2y} 0 dx, &  0\le x \le 5\end{cases}$ = $F(y)=\begin{cases} 0, & -5\le x \le 0\\ 0  ,&  0\le x \le 5\end{cases}$. 
so this is a constant function.
But if we have a joint (continuous) probability density function = 1 such that $F(y)=\begin{cases} \int_0^{2y+10} 1 dx,& -5\le x \le 0\\ \int_0^{10-2y} 1 dx, &  0\le x \le 5\end{cases}$ = $F(y)=\begin{cases} 2y+10, & -5\le x \le 0\\ 10-2y  ,&  0\le x \le 5\end{cases}$
so this is a increasing and then decreasing function. 
Different joint (continuous) probability density gives different characteristics of the functions, increasing, decreasing, and constant...Therefore, without more information, we cannot classify whether it is increasing, decreasing or constant.
Is this right? Is this enough?
A: To make the problem easier, let make the triangle be three points at (0,1), (0,-1), and (2,0)...
In this case... a joint (continuous) probability density function = 1/2 such that F(y)=\begin{cases} \int_0^{2y+2} 1/2 dx,& -1\le x \le 0\\ \int_0^{2-2y} 1/2 dx, &  0\le x \le 1\end{cases} ==> F(y)=\begin{cases} y+1, & -1\le x \le 0\\ 1-y  ,&  0\le x \le 1\end{cases}
so this is a increasing and then decreasing function. This example is right?
Then, this falls into category: neither an increasing nor decreasing function...so I guess my next step is to find a function that is increasing function or decreasing function. ....now I am wondering do they exist ...even they exist, I also have to be able to integrate the function to 1 which is very difficult?....If they exist, can you think of a good way to explain it other than finding an example 
, assuming finding a increasing function example is difficult? If not, do you have any idea or suggestion how to find an increasing or decreasing function?
A: Just realize something .... When the marginal density function begins at y=-1, it must increase at some point before y=1 since the PDF has to be able to integrate to 1 ..on the other than when it got to y=1 .....it must decrease back to 0...so the marginal density function f(y) could not be increasing or decreasing function right...therefore, the answer would be neither increasing nor decreasing 
