Message from discussion Somerhing´s not correct

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Date: Thu, 02 Jul 2009 02:03:10 +0100
From: Henry <s...@btinternet.com>
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Luis A. Afonso wrote:
> Something´s not correct
> 
> San josé State University:
> www.applet-magic.com/samplemin.htm
> it can be read:
> ******** If p(x) is the probability density function for a random variable x, let P(x) be the cumulative probability function; i.e.,
> P(x) = Intg(-infinity, x).p(z)dz
> The probability that the minimum of a sample of size n is x is given by
> Prob.(min size n sample) = n[1- P(x)]^(n-1) * p(x)
> *********************************************
> My comment
> Suppose, for example,
> Uniform [0, 1] Distribution, P(x) = x, p(x) =1
> Prob.(min size n sample) = 21[1- x]^20 * 1
> If x=0.05 then results 7.528,
> This is, no doubt, a remarkable thing (!!!). The Applet is so magic that it not available yet. Please, remove it!
> The writer did intend to say:
> ____Prob (min <=x) = 1- (1-F(x))^n
> ____1- (1-0.05)^21=0.659438______

For a continuous random variable, the probability that the minimum of a 
sample of size n is exactly x is given by 0.

What the author probably intended to say was that the probability 
*density* of the minimum of a sample of size n is x is given by
f(min size n sample) = n[1- P(x)]^(n-1) * p(x)

There is nothing remarkable about a probability density being more than 
1.  Note that the integral of f(x) dx with P(x) = x, p(x)=1 from x=0 to 
0.05 is indeed about 0.659438..., which hardly needs 8000000 runs round 
a simulation to calculate.