Let $X$ and $Y$ be continuous random variables with joint pdf of the form $f(x, y) = c(x+y)$ $0 < x < y < 2$ and zero otherwise.
a. Find c so that f(x, y) is a joint pdf.
I answered this question by double integrals set equal to one and found $c$ = $1 \over 4$
b. Find the marginal pdf’s.
I am unsure if I computed both of these correctly. I am struggling with the boundaries
$$f(x)= \int_x^2 \frac 1 4 (x+y) \, \mathrm{d}y= -\frac {3} 8x^2+\frac 12x + \frac 12$$
$$f(y)=\int_0^y \frac 1 4(x+y) \, \mathrm{d}x=\frac 3 8y^2$$
Can anyone confirm my marginal pdfs?
c. Find the joint CDF.
This is where I am totally lost. I would appreciate a detailed explanation on how to find a joint CDF. I have watched some YouTube Videos and looked at some similar questions here and just am not sure where to start to set-up all of the double integrals and boundaries.
Best Answer
I think it's OK. As a check, both $f(x)$ and $f(y)$ should be $\geq 0$ in their domain and integrate to $1$ (and they do).
As to the cdf, remember that by definition $F(x,y)=P(X<x,Y<y)=\int_{-\infty}^y\int_{-\infty}^x f(u,v)dudv$...
In thi case, you have to integrate in the area given by the intersection of the triangular domain where $f(x,y)$ is defined and the botto-left quarter identified by $(x,y)$...