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Suppose X and Y are Chi-square random variables, and a is some

constant greater than 0. X and Y are independent, but not identically distributed (they have different DOFs).

I want to find

P(X>a,X-Y>0). So I use Bayes' theorem to write

P(X>a,X-Y>0)

=P(X>a | X-Y > 0)*P(X-Y>0)

=P(X>a| X>Y)*P(X>Y)

Now I have an expression for P(X>a) and P(X>Y), but I am at a

loss as to how to evaluate the conditional distribution P(X>a|

X>Y).

I figured out that if Y was a constant (rather than a random variable), then I could write

P(X>a| X>Y) = { 1 if Y>a

{ P(X>a)/P(X>Y) if Y<a

But this does not help evalaute the distribution because I requires knowledge of the value of random variable Y.

I also tried to write

P(X>a,X-Y>0)

=P(X-Y > 0|X>a)*P(X>a)

=P(X>Y| X>a)*P(X>a)

So to evaluate P(X>Y| X>a) I write

P(X>Y| X>a) = int(a...inf (int(0...x f_XY)) dYdX

But this gives some ugly expression which I cannot relate to simply P(X>Y) or P(X>a)

Any help will be much appreciated.