How do I express for the conditional probability and expectation of a binary variable X that depends on the inequality between two other exponentially distributed random variables Y (with rate param $\mu_{i}$) and Z (with rate param $\mu_{j}$)? I am interested in only the mean of the count on successes, what would the expression for the conditional expectation appear like?
$$P(Y<Z)$$ has already been expressed for here
Attempt
Since X obeys a binomial distribution, the pmf for $k$ successes in $n$ trials is given by
$$P(X=k) = \binom{n}{k} p^kq^{n-k} $$
and the expectation is known from here as $$E[X] = np$$. I express the conditional probability as $$P(X|Y<Z) = \frac{P(X=k)P(Y<Z)}{P(Y<Z)}$$. $$F_{X=k|Y<Z}(x) = P(X|Y<Z) = \frac{\sum_{i=1}^{x} \binom{n}{i} p^{i}(1-p)^{n-i} \frac{\mu_{i}}{\mu_{i}+\mu_{j}}}{\frac{\mu_{i}}{\mu_{i}+\mu_{j}}}$$.
Am I doing the right thing here mathematically because I get stuck going ahead. The expectation would appear like $$E[X|Y<Z]$$ but I still cannot get the a final expression for it.
I would be really grateful for any help here. Thx in advance.