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I simply cannot understand this. My TAs aren't available until tomorrow and I really do not want to put this off until then. I'd like to have some idea of how to do this beforehand.

The question is:

The joint density function of X and Y is

f(x, y) =

2
for 0 < x < y, 0 < y < 1

0 otherwise

a) Compute the density of X. b) Compute the density of Y. c) Are X and Y independent?

Integrals have something to do with this, I know, but I'm not sure how to use them to get the answers I want. Independence in terms of probability I know but I don't know how to apply it to this problem.

TLDR: I need an explanation (thorough, please) on how to approach this problem so I know how to complete it and obtain the correct answer.

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Okay. When you have a joint density function f(x,y) and you want to find the density of x, and lets say this g(x), then integrate f(x,y) with respect of y. Similarly, if you want to find the density of y, call this h(y), integrate f(x,y) with respect of x. Then, once you get these two densities, try multiplying them together, i.e. g(x)h(y), and if this is equal to f(x,y) then they are independent. Hope this helps.

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  • $\begingroup$ Is it really that simple? And, just to clarify, when I integrate with respect to y I am treating as a x constant, correct? It's been a while since I've been in a Calculus class. $\endgroup$ – Super Rhinocerus Apr 27 '15 at 22:12
  • $\begingroup$ Yup its really that simple. You just have to watch out for your domains. And yes, treat x as a constant when integrating/differentiating y, vice versa. [I am majoring in Maths and Stats] $\endgroup$ – Skipe Apr 27 '15 at 22:43
  • $\begingroup$ After looking at this problem a bit more, I'm confused how to set up the integral given the parameters. We only care about the integral where it's not 0, right? So the bounds for x would be from 0 to y and the bounds for y would be from 0 to 1? $\endgroup$ – Super Rhinocerus Apr 27 '15 at 22:53
  • $\begingroup$ If you are "integrating out" $x$, then indeed $x$ goes from $0$ to $y$. If you are integrating out $y$, then $y$ goes from $x$ to $1$. I think the best way to see this is to draw a picture. The joint density "lives" on the triangle with vertices $(0,0)$, $(1,1)$, and $(0,1)$. Draw that triangle. $\endgroup$ – André Nicolas Apr 27 '15 at 23:12
  • $\begingroup$ I can see why x goes from 0 to y, but why does y go from x to 1? Doesn't it go from 0 to 1? $\endgroup$ – Super Rhinocerus Apr 27 '15 at 23:19

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