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Suppose that the joint distribution of $X$ and $Y$ is uniform over the region in the $xy$-plane bounded by $x=-1,x=1,y=x+1, \text{ and }y=x-1$.

What is $\mathbb{P}(XY>0)$?

What is the conditional p.d.f. of $Y$ given that $X=x$?

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What did you try? Where are you stuck? Did you draw a picture of the region of interest? If you did, you probably saw that $XY\gt0$ with probability $\frac34$. – Did Jun 24 '12 at 8:14
sorry,I just stuck in the second part,the first question is a lead-in. – perry zhu Jun 24 '12 at 9:27
up vote 3 down vote accepted

The region in the $XY$-plane is as shown below.

enter image description here

HINT for the first part. Identify the regions where $XY > 0$. And integrate over the region to get $\mathbb{P}(XY > 0)$.

HINT for the second part. Recall that $f_{Y|X=x} = \dfrac{f_{XY}}{f_X}$, where $f_X = \displaystyle \int_y f_{XY} dy$.

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+1 for diagram. I am not sure I would use calculus for the first part, rather than calculating the areas from the diagram. For the second part, inspection should give the answer more quickly. – Henry Jun 24 '12 at 8:47
thank you a lot. – perry zhu Jun 24 '12 at 9:30

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