> 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.
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