Why my solution is not true?

Question

We know that $X_1,X_2$ and $X_3$ are three independent exponential random variables, with rates $\lambda_1,\lambda_2$, $\lambda_3$ respectively. We want to calculate $$E[\max{X_i}| X_1<X_2<X_3]$$ The answer is equal to $$\frac{1}{\lambda_3}+\frac{1}{\lambda_2+\lambda_3}+\frac{1}{\lambda_1+\lambda_2+\lambda_3}$$ I have read the solution for this problem, and it seems to be correct for me, but I cannot understand why we cannot explain the following solution for this problem:

As we know that $X_1<X_2<X_3$, so $\max{X_i}=X_3$, hence we have $E[\max{X_i}| X_1<X_2<X_3]=E[X_3]=\frac{1}{\lambda_3}$

Any help?

Answer 1

Following your logic rigorously, one gets the correct identity $$E[\max{X_i}\mid X_1<X_2<X_3]=E[X_3\mid X_1<X_2<X_3].$$ Note that the RHS is definitely not $E[X_3]$.

Answer 2

Intuitively speaking, the solution you suggest simply ignores the constraint $X_1<X_2<X_3$. Yes, you calculate the expected value of $X_3$, but you include in your calculation events where $X_3$ is not the maximum.