# Thread: one last joint distribution

1. ## one last joint distribution

joint density function of X and Y:

$\displaystyle f(x,y)=\left\{ \begin{array}{lr} k(x+y)&0\le x+y\le 1; x,y\ge 0\\ 0&otherwise \end{array} \right.$

If I was looking for k, would $\displaystyle \int_0^{1-y} \int_0^{1-x} k(x+y) dy dx=1$ be the right way to start?

2. NO
The outer integral cannot contain variables...
see my change below

$\displaystyle \int_0^{1} \int_0^{1-x} k(x+y) dy dx=1$ is the right way to start

3. When it comes to finding the marginal densities, would I then use 1-x and 1-y as the upper limits?

4. no, review calc 3
IF you had variables on that outer integral HOW can you end up with a constant?

5. I understand that about finding the constant. What I mean is, once i have found the constant (3 I think), and want to find the marginal densities, would I then use:

$\displaystyle f_Y(y)=\int_0^{1-y} f(x,y) dx$

and

$\displaystyle f_X(x)=\int_0^{1-x} f(x,y) dy$

or do I have to stick with the initial values?

6. These look ok

Originally Posted by billym
I understand that about finding the constant. What I mean is, once i have found the constant (3 I think), and want to find the marginal densities, would I then use:

$\displaystyle f_Y(y)=\int_0^{1-y} f(x,y) dx$

and

$\displaystyle f_X(x)=\int_0^{1-x} f(x,y) dy$

or do I have to stick with the initial values?