Joint probability density

[anon_econ]

If the joint probability density of X and Y is given by

f(x,y)= 2 for x>0, y>0, x+y<1
          0 elsewhere,

find P(X>2Y).

[priyanka]

ans P(X>2Y)= 1/3

draw the following area
x>0, Y>0,x+y<1, x>2y
now try to see that when x goes from 0 to 2/3, y goes from 0 to x/2
when x goes from 2/3 to 1, y goes from 0 to 1-x
now, use these limits to find out required probability (by integrating the pdf)

[anon_econ]

Got it..thanks a lot!

[ritu]

hey vasudha can u explain it again....i dint get it...

[anon_econ]

Ritu look at the examples given in freund. I also simply jumped to this question w/o studying joint pdfs first..don't do that. U'll get it only if u have studied joint pdfs. Ask me later if u don't get it.

[ritu]

yes vasudha pls explain...i tried...

[anon_econ]

It's very difficult to explain this w/o using a graph, but i'll try.
First of all plot the domain of the pdf, that is the area over which if u integrate, u will get 1. This is the area enclosed by x>0, y>0, x+y<1 (bcoz the value of the pdf is 0 everywhere else).
Now in the same figure, u identify the area that u want, that is x>2y. U get a triangle. Now if v 'integrate the pdf over this triangle', we will get the probability that (x,y) lies in this region (which is what we want).
The only trick is to figure out the limits when we 'integrate over the triangle'.
What we want is that every point in the triangle should be covered. If i take any particular x b/w 0 & 2/3, all the corresponding values of y in the region vary from 0 to x/2. So to 'cover all values of y for a single value of x', i integrate '2'  wrt y over 0 to 2/3. Now i want the same thing for all values of x. So i integrate wrt x from 0 to 2/3.
This gives me a part of the triangle. Now for the other part, i proceed similarly. For a particular x b/w 2/3 & 1, y varies b/w 0 and 1-x. So integrate wrt y from 0 to 1-x. We need this for all the values of x, so now integrate wrt x from 2/3 to 1.
Add the 2 integrals.
I hope u got the hang of it. Look at example 3.16 on pg 97 again now.

[ritu]

hey i got it now.....i could visualize ur explanation...thanx soo much..:)