Joint/Conditional distribution

[OFDM]

I'm having a problem evaluating a distribution-

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.

 

[kurt.physics]

The best approach in this case is to use the law of total probability. This states that the probability of an event A is equal to the sum of the probability of the event A given each of the possible values of a random variable. Using this law, we can write:P(X > a, X - Y > 0) = $\sum_{y=0}^{\infty}\ P(X>a\ |\ X-Y=y)\ P(X-Y=y)$Now, for each value of y, we can calculate the probability of X > a given that X - Y = y. We can then multiply this by the probability that X - Y = y and add it to the sum. This approach will allow us to evaluate the distribution without needing knowledge of the value of Y.