Suppose random variables X, Y have joint probability density function $f(x, y)$. How do i find the marginal probability density function of X , Y if the support is

$$ \begin{cases} 0 < x < 1 \\ x < y < 1 + x \end{cases} $$

I know that I need to integrate with respect to X to find P.D.F. of Y and vice versa. But I don't know what the boundary of the integrals should be. Thanks in advance.


To solve this problem, I need to draw a picture. I strongly recommend that you do so also.

Note that since $0\lt x\lt 1$, we have $0\lt y\lt 2$.

So draw the rectangle with corners $(0,0)$, $(1,0)$, $(1,2)$, and $(0,2)$.

Draw the lines $y=x$ and $y=x+1$.

Our random variable lives in the rectangle, and between these two lines.

Now finding the (marginal) density function of $X$ is easy. We have to "integrate out" $y$. So $y$ will travel from $x$ to $x+1$.

In principle, finding the density function of $Y$ is also easy, we have to integrate out $x$.

But if you look at the picture, you can see that we will have to break up the integral into two parts.

If $0\lt y\le 1$, we will be integrating from $x=0$ to $x=y$. From $1\lt y\le 2$, we will be integrating from $x=y-1$ to $x=1$.


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