Help: calculate the mean of maximum of multiple Gaussian r.v

[changfa]

I need to calculate the mean of maximum of K i.i.d. Gaussian r.v.s., each with N(mu, sigma^2).

let M_K denote max(x1,x2,...,xK) and it is straight forward to show that the CDF of M_K is (F(x))^K where F(x) is the CDF of N(mu, sigma^2). Then the pdf of M_K is given by K*(F(x))^(K-1)*f(x).

The final step is to calculate the mean value of M_K as
E{M_K} = \int_{0}^{+oo} x * K*(F(x))^(K-1)*f(x) dx.


Does anyone know if there is close-form expression for this? or how to numerically calculate this integral in Matlab? (I tried but failed)

Thanks a lot!\]

 

[steve10]

Re: Help: calculate the mean of maximum of multiple Gaussian

This can be done (a close form) for K=2 easily, and I doubt there is a way to get it for K>2. By the way, shouldn't the lower limit of the integration be -∞?

 

[changfa]

Re: Help: calculate the mean of maximum of multiple Gaussian

The random variables for my case have physical meaning and are positive, therefore, I think the lower limit of the integral is -oo.