Thread: probability question

X and Y are independent random variables. If P(X>u)>=P(Y>U) for all u, then P(X>Y)>=1/2

Thanks a lot

Originally Posted by violetsf

X and Y are independent random variables. If P(X>u)>=P(Y>U) for all u, then P(X>Y)>=1/2

Thanks a lot

Assume X and Y are continuous random variables with pdf f(x) and g(y) (the argument is easily modified for the discrete case). Then the joint pdf is f(x) g(y) due to independence of X and Y.

What my argument boils down to is that in the XY-plane, the 'weighted area' below the line Y = X is greater that the 'weighted area' above the line Y = X since :


1. .

2. .

3. .

Now use 1., 2., 3. and to show that .

Thank you very much!
how about #3, I mean where is P(X=Y)

Originally Posted by violetsf

Thank you very much!
how about #3, I mean where is P(X=Y)

Obviously you just replace the typos < and > with and .

Another question: f(u) and g(u) are not the same, right?
then how can you compare #1 and #2 ?
Thanks!

Originally Posted by violetsf

Another question: f(u) and g(u) are not the same, right?

[snip]

Right.

Originally Posted by violetsf

then how can you compare #1 and #2 ?
Thanks!

I'll need to think about this! It seems less simple now than when I posted.

I'm confident the approach is correct, but it might not be the most fruitful approach (I don't see what other approach could be taken at this moment).