Conditional expected value of the maximum of n normally distributed variables What is the conditional expected value of the maximum of $n$ normally distributed variables $x_i$,  conditionally on exceeding some threshold $z$? 
That is, for $x=\max(x_1,...,x_n)$, the probability that $x$ will exceed some threshold $z$ is $P[x>z]=1-F[z]^n$. For large $n$, the distribution of $x$ can be approximated by the Gumbel distribution, and the expected value of $x$ asymptotically equals 
$$E[x]=\mu + \gamma \sigma (2 \log(n))^{1/2}$$ 
Does there exist any such basic approximation formula, when $x_i$ are normally distributed and the exact value of $z$ is known?

$$E(x\mid x\gt z)=z+\frac1{1-F(z)^n}\int_z^\infty(1-F(t)^n)\mathrm dt$$
