I am struggling with determining bounds of a quadruple integral and I hope that maybe someone might be able to help me out.
I am looking into a situation where I have four independent, exponentially distributed variables, let's say, A, B, C and D. I would like to calculate the probability
$ P(B < A < C) P (B < D) $
In order to calculate this, I think, I need to integrate over the joint density of the 4 variables, which, since all of them are independent of each other, is simply a product of the separate univariate density functions. I am struggling to define the limits of the integral though. I would be fine if I had to deal with the first probability only, but the need to account for the second probability confuses me. How do I define the integral bounds for the variable B so that both of the inequalities are met?
Here is my idea for now, although I am pretty sure it is not correct:
$ \int^{\infty}_0 f(d) \int_0^d f(b) \int_b^{\infty} f(c) \int_b^c f(a) da dc db dd$
I would appreciate any hints. Thank you.
Your integral would be the correct answer for probability $$ \mathbb P(B<A<C, B<D) $$ And $\mathbb P(B<A<C, B<D) \neq \mathbb P(B < A < C)\cdot \mathbb P (B < D)$ since these events are dependent. So calculate each of these two probabilities separately and multiply. $$ \mathbb P(B < A < C) = \int^{\infty}_0 f(b) \int_b^\infty f(a) \int_a^{\infty} f(c) \,dc\, da\, db $$ and $$ \mathbb P(B < D) = \int^{\infty}_0 f(b) \int_b^\infty f(d)\, dd\, db. $$