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If I have a joint density function for X and Y:

$f_{X,Y}(x,y) = \begin{cases} \pi x \cos(\frac {\pi y} 2) & 0 \le x \le 1, 0 \le y \le 1 \\ 0 & \text{otherwise} \\ \end{cases}$

How do I find the marginal density function for X?

I think I need to integrate $f_{X,Y}(x,y)$ over $dy$ but what do I integrate it from? Should it be 0 and 1 or 0 and x or x and 0? I've looked over a lot of examples with different domains but I can't figure out their method in getting the range for integration. They always skip that step in the working because apparently it should be obvious, but I can't figure it out. Can anyone help me please?

Mhenni Benghorbal
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Jigglypuff
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  • From zero to one. – Eckhard Apr 09 '13 at 19:17
  • In principal you integrate from $-\infty$ to $\infty$, but since the joint density vanishes outside the interval $0\le y \le 1$, it is sufficient to integrate from $0$ to $1$, as Eckhard pointed out correctly. – Matt L. Apr 09 '13 at 19:20
  • What would be the integral limit if condition was $0 \lt x \lt y \lt 1$ ? – Taylor Aug 03 '17 at 04:34

1 Answers1

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The marginal density is given by $$ f_X(x)=\int_{-\infty}^\infty f_{X,Y}(x,y)\,\mathrm dy,\quad x\in\mathbb{R}. $$ Now, this equals $$ \int_{0}^1 \pi x\cos\left(\frac{\pi y}{2}\right)\,\mathrm dy,\quad \text{if }\;0\leq x\leq 1 $$ and $0$ otherwise.

Stefan Hansen
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