[R] probability density function for maximum values in repea

[(Ted Harding)]

On 28-Sep-09 12:15:39, Joerg van den Hoff wrote:
> this is probably not really a R specific question, if so apologies
> for off-topic posting:
> 
> I'm interested in the probability density function of the maximum
> values from repeated samples of size N from a normal distribution:
> 
> smp <- rnorm(N, meanval, stdev)
> 
> with some mean 'meanval' and standard deviation 'stdev'.
> 
> I would like to know what is the frequency distribution of max(smp) if
> I draw many such
> samples?
> 
> if I investigate this simply via a simulation, I get of course
> approximate
> results (and see that the resulting distribution is not quite normal
> anymore, that the mean increases with increasing N, etc.).
> 
> my question: does somebody know whether there exists an analytical
> expression for the distribution of max(smp) (or where to look)?
> 
> thanks,
> joerg

Let Pmax(x,N) be the probability that the maximum of N is <= x.

Pmax(x,N) = Prob(all N values <= x) = (Prob(a single value <= x))^N
          = (pnorm(x,meanval,stdev))^N

Hence the frequency distribution is the derivative of this with
respect to x, say pmax(x,N):

pmax(x,N) = N*((pnorm(x,meanval,stdev))^(N-1))*dnorm(x,meanval,stdev)

Ted.

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Date: 28-Sep-09                                       Time: 13:45:49
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