Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 1 down vote accepted

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 $x=1$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.