Isn't this just the "law of total expectation"?
I don't really understand your point I'm afraid.
EDIT: Yes, we can also go by the second approach.
Then how to calculate E[B.I(A<B<C)] exactly ? (I'm not sure how to start with this integral.)
First a remark: we may assume that A, B and C are independent random variables (this may help a lot).
E [ B | A < B < C ] = \int_{0}^\infty E [ B \, | \, A < B < C, A = a ] \; f_{A} ( a ) \; da
= \int_{0}^\infty \int_{0}^\infty E [ B \, | \, A < B < C, A = a, C=c ] \; f_{A} ( a ) \...
I've been struggling with this problem for more than 4 days now:
Let A, B and C be exponential distributed random variables with parameters lambda_A, lambda_B and lambda_C, respectively.
Calculate E [ B | A < B < C ] in terms of the lambda's.
I always seem get an integral which is...