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Re: AI-GEOSTATS: Standard deviation, Variance

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  • Ruben Roa
    Isobel did not write the careless paragraph about the central limit theorem (CLT) Don replied to, as pointed out by Digby. I wish to add something to what Don
    Message 1 of 9 , Dec 5, 2002
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      Isobel did not write the careless paragraph about the central limit theorem
      (CLT) Don replied to, as pointed out by Digby. I wish to add something to
      what Don said about the conditions under which the CLT applies, and that
      people usually miss in considering the universality of the CLT. See below.

      >> Let X_1,...., X_n be a sequence of independent,
      >> identically distributed
      >> random variables with common mean m and common
      >> standard deviation
      >> sigma. Let Z_n be defined as a normalized sum
      >>
      >> Z_n = [S_n - m]/ (sigma/sqt root of n),
      >> S_n = [Z_1
      >> +.....+ X_n]/n
      >>
      >> S_n is the sample mean
      >>
      >> Let F_n(z) be the cumulative probability
      >> distribution function for Z_n
      >> and let G(z) be the cumulative probability
      >> distribution function for the
      >> standard Normal,. Then F_n(z) --> G(z) as n
      >> increases.
      >>
      >> Note two things about this statement, (1) the
      >> theorem does not say how
      >> "fast" the cdf for Z_n approaches the standard
      >> Normal, (2) the speed of
      >> convergence depends on z. Also the speed of
      >> convergence depends on the
      >> distribution type of the X_i's

      Note also the sum operation. The CLT, more precisely called the Additive
      CLT, applies to sums of pairwise independent random variables as n tends to
      infinity. But if the operation is multiplication with equal-signed r.v.,
      then convergence in distribution is towards the lognormal, not the normal.
      It might well be that when considering natural phenomena, multiplicative
      processes be more or equally common than additive ones, as we oftenly
      observed skewed continuous data.
      Rubén
      http://webmail.udec.cl

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    • Isobel Clark
      Thanks to Rubén and Digby for pointing out what I had misunderstood about Don Myers email. It had not occurred to me (duh) that the lines starting would
      Message 2 of 9 , Dec 6, 2002
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        Thanks to Rubén and Digby for pointing out what I had
        misunderstood about Don Myers' email.

        It had not occurred to me (duh) that the lines
        starting '>' would be read as being from me rather
        than part of a forwarded email.

        Another score on the dumb side. Apologies for the
        strong reaction to Don's email if (on this occasion)
        he was not criticising my contribution.

        Isobel

        http://uk.geocities.com/drisobelclark


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