How to compute Kendall tau when X and Y are dependent? I am stuck in the following problem.
Let ($X_1$,$Y_1$) and ($X_2$,$Y_2$) be independent and identically distributed continuous bivariate random variables with joint probability density function:
$f(x,y) = \exp(-y)$ when $0< x < y< \infty; =0,$ elsewhere.
I do not know how to proceed to get $P(X_2>X_1, Y_2>Y_1)\ +\ P(X_2 < X_1, Y_2$< $Y_1$) as a part of
calculating Kendall's tau. 
If anyone knows, please let me know how to express each of the above two probabilities in a form of integration. 
 A: Consider  $P(X_2>X_1, Y_2>Y_1)$ and note that the first and second points are independent so the joint distribution is the product of two two bivariates.
Below is a collection of points from the bivariate density in the question (in grey) to illustrate the density and clarify what's going on in the integral. 
The red point represents a particular instance of $(x_1,y_1)$ and the blue point represents $(x_2,y_2)$ such that the joint inequality in the first probability is the region above and to the right of the red point (i.e. the values covered by the inner double integral). 
The red point then ranges over the whole set of values for $(x_1,y_1)$ (outer double integral). 

This is perhaps the most basic way of doing it.
[There are other approaches you could consider, which may be simpler.]

The green point (below and left of the red one) could be used to visualize the integration for  the other probability, but you can just invoke symmetry to interchange the roles of the first and second point which should simplify things, essentially giving you the second one for free.
