- 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,

Note also the sum operation. The CLT, more precisely called the Additive

>> 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

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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* Support to the list is provided at http://www.ai-geostats.org - 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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