Hi, I have a problem that I'm not able to solve because of my
non-statistics background.
Assume we do a Monte Carlo experiment with M=1000 samples, each of which
has N=100 data drawing from a p.d.f. p(x) of normal distribution.
For each sample, we first sort the 100 data in an increasing order X1 <=
X2 <= ... <= X99 <= X100, and calculate the probability that a random
datum obeying p(x) is located between minus infinity and X1, that is,
the integral INT_{-infinity}^{X1} p(x) dx. If we calculate the mean of
this probability for M=1000 samples, then the mean will approach
1/(N+1)=1/101. That is to say, the expectation of this probability is
1/(N+1).
My Monte Carlo experiment showes that the above is very likely correct.
Could anyone mathematically prove whether the above is true?
Furthermore, if we replace the 1st minimum value X1 with nth minimum
value Xn, then the expectation of the probability will be n/(N+1). If
it is true for a normal distribution function, is it also true for any
other p.d.f. p(x). It seems to me that the guess is also true for a
uniform distribution.
Thank you in advance.
--
Qiang Li, Ph.D.
Department of Radiology
The University of Chicago
Tel: (773) 834-5096
Fax: (773) 702-0371
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