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______ Simulation (data 8E6 items) gave 0.660, 0.660. Prob (absolute deviation > 0.0005 | N=8E6) <= 2*EXP(-2*8E6*0.25*1E-6) = 2*EXP(-4) = 0.037= 3.7% (According to Dvorestky - Kiefer - Wolfowitz inequality, states). REM "ckontas" CLS DEFDBL A-Z DIM x(22) INPUT " x= "; x0 p = 1 - (1 - x0) ^ 21 PRINT USING "#.### "; p all = 8000000 RANDOMIZE TIMER FOR j = 1 TO all LOCATE 4, 4: PRINT USING "########"; all - j minor = 8 FOR i = 1 TO 21 x = RND IF x < minor THEN minor = x NEXT i IF minor > x0 THEN GOTO 10 okk = okk + 1 10 NEXT j: LOCATE 10, 10 PRINT USING "#.### "; okk / all END
> 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.
******** 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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My response
__1__You agree, it´s evident, that the *genius* that posted such a thing should be severely adverted by the University. __2__You are sufficiently smart to understand that the simulation is an addendum to the formula, or not? __3__Your opinion what the author *intended to say* is a simple and unintelligent extrapolation. Doing so you think, in your *naivety*, people will to absolve him. Furthermore it was him (not me) that gave the formula of the probability of the minimum be x: you should correct him as long as he published the post. __4__Aspiring you to be known as a scrupulous person in Statistics, why you didn’t correct the error your fellow barbarous countryman got?. That was YOUR task.
On 2 July, 04:09, "Luis A. Afonso" <lic...@hotmail.com> wrote:
> __3__Your opinion what the author *intended to say* is a simple and unintelligent extrapolation.
You are correct to say it is simple, but wrong to say unintelligent or extrapolation, since it only involves inserting a single word used in the same context on the same page, rather than integrating a formula.
Sometimes you worry too much about others, and your ability to be polite often seems to be a couple of slices of bread short of a sandwich. Your "Somerhing" was not a word, but I am not at all concerned about it, because it is obviously a typing error replacing t by the adjacent r.
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