Stopping time

May 2010
1
0
Let (X1n,X2n), n=1,2,..., be iid, each (X1n,X2n) being independent Exponential (1) and Exponential (2) respectively. Fix a t > 0 and define the stopping time

N = min{n>1| X1n>t, X2n>t}

Obtain the joint d.f. F(x1,x2) = P(X1n < x1, X2n < x2 ) and hence the joint pdf f(x1,x2) of the stopped pair (X1n,X2n)

I've tried something, but I can't finish it. Here is what i've done:

sum from n=1 to infinity ( P(X1n<x1, X2n<x2, N=n)
sum from n=1 to infinity ( P(x1n<x1, x2n<x2, X1n>t, X2n>t, X1i<t, X2i<t, 1<i<n-1)
sum from n=1 to infinity ( P(X1i<t, X2i<t)^n-1 * P(x1n<x1, x2n<x2, X1n>t, X2n>t))

From this point, I am a little bit confused on how to proceed to evalute these probabilities!

A little help would be very appreciated!!
 
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