# Interpreting the Marginal Probability Density Function of a Rectangle

the common probability density function $$f(x,y)$$ of two random variables $$X,Y$$ is constant in the rectangle $$Q$$ and vanishes outside of it.

$$f(x,y)=\begin{cases}\frac{1}{2} &\quad, (x,y)\in Q\\0 &\quad, \text{else}\end{cases}$$ Now I'm interested in the marginal probability density function for $$X$$. We have to consider two cases.

Case 1: $$-1\leq x \leq 0$$

$$f_X(x)=\int_{-\infty}^\infty f(x,y) dy = \int_{-1-x}^{1+x} \frac{1}{2} dy = \dots = 1+x$$

Case 1: $$0\leq x \leq 1$$

$$f_X(x)=\int_{-\infty}^\infty f(x,y) dy = \int_{-1+x}^{1-x} \frac{1}{2} dy = \dots = 1-x$$

So we get

$$f_X(x)=\begin{cases} 1+x &\quad, -1\leq x \leq 0\\ 1-x &\quad, 0\leq x \leq 1 \\ 0 &\quad, \text{else}\end{cases}$$

Now I kind of have a hard time interpreting $$f_X$$. Can someone try to explain why the result we got makes sense?

• Your work and answer are correct - what is your question exactly? $f_X$ is a density function, not sure what there is to "interpret" here. – Math1000 Aug 1 at 13:13
• Like I think I should be able to look at $f_X$ and think "Yeah, that is a result which does indeed make sense for $f$". E.g. before getting $f_X$ you might think about what kind of solution you get and then you get something and you aren't really surprised. I can't do that here, I don't see why one would expect this kind of $f_X$. – handy Aug 3 at 11:35