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.


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 $(x,y)$...


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