Domain in marginal density functions I am so confused about the domain for marginal density of this problem...
Here is the joint density function :

$$
f(y_1,y_2) = \begin{cases} e^{-y_1} & 0\le y_2 \le y_1 \le \infty \\ 0 & \text{elsewhere} \end{cases}
$$
Here is the solution:

I perfectly understand how we get those two marginal densities. What I don't understand is the domain for these two densities.
Can someone explain please?

The support for the joint density function, $\{(Y_1,Y_2):0\leq Y_2\leq Y_1 < \infty\}$, can be expressed as either: 


*

*$\{\, (Y_1,Y_2) : Y_1\in [0,\infty), Y_2\in [0,Y_1] \,\}$

*$\{\, (Y_1,Y_2) : Y_2\in [0,\infty), Y_1\in [Y_2,\infty) \,\}$ 


Hence the limits for the marginal density functions.

To put it another way, to find the marginal density $f_1$ at $Y_1=y_1$, integrate over the support for $Y_2$ at that value, which is $0\leq Y_2\leq y_1$.
Likewise, to find the marginal density $f_2$ at $Y_2=y_2$, do so over the integrate over the support for $Y_1$ at that value, which is $y_2\leq Y_1 < \infty$.
