suppose that $x_1,x_2,x_3$ are independently drawn according to a CDF $F$.
I understand the mean of highest or second highest value can be found using the order statistics.
My question is if we know that $x_1$ and $x_2$ are the top two highest values and if we know the exact value of $x_2$, then what would be the expected value of $x_1$?
Any hint or help will be highly appreciated!
We have $\mathsf P(X_1\le x_1)=F(x_1)^3$, thus for $x_1\ge x_2$
$$ \mathsf P(X_1\le x_1\mid X_2=x_2)=\mathsf P(X_1\le x_1\mid X_1\ge x_2)=\frac{F(x_1)^3-F(x_2)^3}{1-F(x_2)^3} $$
and thus
$$ \mathsf E(X_1\mid X_2=x_2)=x_2+\int_{x_2}^\infty\left(1-\mathrm P(X_1\le x_1\mid X_2=x_2)\right)\mathrm dx_1=x_2+\int_{x_2}^\infty\frac{1-F(x_1)^3}{1-F(x_2)^3}\,\mathrm dx_1\;. $$