1
$\begingroup$

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 $X=x$?

  • 0
    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 $\frac34$.2012-06-24
  • 0
    sorry,I just stuck in the second part,the first question is a lead-in.2012-06-24

1 Answers 1

3

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 $f_X = \displaystyle \int_y f_{XY} dy$.

  • 1
    +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.2012-06-24