I find this fascinating. Apparently what I said is almost entirely wrong.
What I said was 'I was taught that.......' I do not recollect Don Myers being in my classrooms as an undergraduate (or during my MSC for that matter). You know, I welcome criticism, especially when I get things wrong. I have a big problem with people who do not actually read what I write but react at some visceral level to what they think I said. Also, I must be really stupid, because the comments given by Don include the statement " If any of the conditions in the theorem are not satisfied then the theorem may not apply. " Which, I am fairly sure, is what I was trying to say. Isobel Clark http://uk.geocities.com/drisobelclark/resume --- "Donald E. Myers" <[EMAIL PROTECTED]> wrote: > Regrettably the following statement by I. Clark is > almost entirely wrong > See below for a correct statement of the CLT, the > problem in part is > simply carelessness in terminology and replacing > correct > statements/formulations by sort of heuristic ones > (which are not correct) > Donald E. Myers > http://www.u.arizona.edu/~donaldm > *********************************************************************** > Isobel Clark wrote: > > >>The reason is simple and comprehensive.... > >> > >>Assume a population with ANY distribution of > >>elements. Then randomly select > >>a number of sample elements from the population to > >>characterize the > >>underlying population. That distribution of sample > >>elements ALWAYS tends > >>toward a normal [Gaussian] distribution. And the > >>mean and standard deviation > >>of the sample distribution are unbiased > >>representations of the mean and > >>standard deviation of the underlying population. > >> > > > > > *************************************************************************** > > CLT > 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 > > If any of the conditions in the theorem are not > satisfied then the > theorem may not apply. The convergence in this > theorem is what is called > "convergence in distribution", this is one of the > weakest forms of > convergence for a sequence of random variables. > There are theorems that > will give estimates or bounds on the speed of > convergence. There are > also special cases of this theorem that are somewhat > simpler such as the > the Normal approximation to the Binomial > > The simplest proof of the theorem above uses > characteristic functions > (Fourier Transforms of the densities). > __________________________________________________ Do You Yahoo!? Everything you'll ever need on one web page from News and Sport to Email and Music Charts http://uk.my.yahoo.com -- * To post a message to the list, send it to [EMAIL PROTECTED] * As a general service to the users, please remember to post a summary of any useful responses to your questions. * To unsubscribe, send an email to [EMAIL PROTECTED] with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list * Support to the list is provided at http://www.ai-geostats.org
