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The Central Limit Theorem

steve_bank

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https://en.wikipedia.org/wiki/Central_limit_theorem


In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution (informally a bell curve) even if the original variables themselves are not normally distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions. This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern general form, this fundamental result in probability theory was precisely stated as late as 1920,[1] thereby serving as a bridge between classical and modern probability theory.


For example, suppose that a sample is obtained containing many observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic mean of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the probability distribution of the average will closely approximate a normal distribution. A simple example of this is that if one flips a coin many times, the probability of getting a given number of heads will approach a normal distribution, with the mean equal to half the total number of flips. At the limit of an infinite number of flips, it will equal a normal distribution.

The central limit theorem has several variants. In its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations, if they comply with certain conditions.

The earliest version of this theorem, that the normal distribution may be used as an approximation to the binomial distribution, is the de Moivre–Laplace theorem.

The CLT is a foundation for practical statistical analysis and sampling methods. It leads to the sampling distribution of the mean.

https://stats.libretexts.org/Booksh..._The_Sampling_Distribution_of_the_Sample_Mean

And confidence intervals

https://en.wikipedia.org/wiki/Confidence_int


Philosophical issues
The principle behind confidence intervals was formulated to provide an answer to the question raised in statistical inference of how to deal with the uncertainty inherent in results derived from data that are themselves only a randomly selected subset of a population. There are other answers, notably that provided by Bayesian inference in the form of credible intervals. Confidence intervals correspond to a chosen rule for determining the confidence bounds, where this rule is essentially determined before any data are obtained, or before an experiment is done. The rule is defined such that over all possible datasets that might be obtained, there is a high probability ("high" is specifically quantified) that the interval determined by the rule will include the true value of the quantity under consideration. The Bayesian approach appears to offer intervals that can, subject to acceptance of an interpretation of "probability" as Bayesian probability, be interpreted as meaning that the specific interval calculated from a given dataset has a particular probability of including the true value, conditional on the data and other information available. The confidence interval approach does not allow this since in this formulation and at this same stage, both the bounds of the interval and the true values are fixed values, and there is no randomness involved. On the other hand, the Bayesian approach is only as valid as the prior probability used in the computation, whereas the confidence interval does not depend on assumptions about the prior probability.


The code takes the average of a sample size through a random vector. Sort asscebding gives a cumulative distribution.


clear
samp_size = 50
N = 10000
Nsamp = N/samp_size
r = grand(N,1,"nor",10,3)
//r = grand(N,1,"exp",10)
rs = gsort(r,'g','i')
index = 0
for i = 1:Nsamp
acc = 0
for j = 1:samp_size
acc = acc + r(index + j )
end
s(i) =acc
index = index + samp_size-1
avg(i) =acc/samp_size
end
avgs = gsort(avg,'g','i')
 

steve_bank

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Exponential histogram and cumulative distribution

expdist.png

expcum.png

Normal cumulative distribution

norpcum.png

Cumulative distribution of sample averages from exponential distribution. Approximatly normal.

expavgcum.png
 

fromderinside

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If there are samples there are ratios determinable. If there are ratios then those can serve as markers of distribution which can be expressed as a function. If there is a function it can serve as an axis for representing the distribution of sample for which a ratio was found. That ratio would be normal for the expressed ratio of samples.

rat tat bumpf
 

fromderinside

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Clarification.

I presume Samples are randomly distributed. Distributions can be partitioned into ratios. Ratios are markers. Ratios can be described as functions. Functions can be represented graphically. Using a ratios of one as a model, it is self evident that ratios can be represented as functions producing sample plots in the classical normal distribution shape.

It seems to me that the central limit theorem can be extended to cover all real number ratios.

Hokay? Hokay.

tat bumpf?
 

steve_bank

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Random means no correlation between variables. A random sample enas the sampling event evet has no statistical effect or correlation to the next sampling event. No matter how many times you toss a die the probabilityThe histogram os a sampe group from a normal distribution will be normal. A histogam of random smaple group from an expnntial distribution will be exponential.

is always one in six on the next toss.

A random vector of an exponential al distribution s not normally distributed, it is exponentially distributed. Sample averages from a randomized vector of an exponential distribution are approximately normally distributed. The Central Limit Theorem.

An important question in sampling theory is how many samples do you need from a population to have a certin comfdence in the sample parameters?

You have 10000 widgets in boxes, how many do you sample to have high confidence in the estimate of a mean or average value such as weight or size?
 

Swammerdami

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One of the more remarkable digressions from this topic is the  Cauchy Distribution. I have attached an image comparing the shapes of a Gaussian distribution and a Cauchy distribution.

At first glance they both seem "bell-shaped", though superimposed like this it is obvious that Cauchy has more kurtosis. But in fact, Cauchy variates are ridiculous: "The Cauchy distribution is often used in statistics as the canonical example of a "pathological" distribution since both its expected value and its variance are undefined."

One way to generate a Cauchy variate is to take the ratio of two zero-mean Gaussian variates. Another way is simply to take the tangent of a uniform variate on (-pi, pi). ... And No, sums of Cauchy variates do NOT obey the "Central Limit Law"!


Cauchy.png
 

Alcoholic Actuary

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When attempting to use CLT in real life, you will spend the majority of your time attempting to establish that your independent random variables are in fact 1. Independent and 2. Random.

aa
 

steve_bank

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You never blindly apply statistics. You first analyze data samples. Then come up with an approch.

My frist step was always to plot a cumulative distribution of sample data. See if it looks like a known distribution

If you have a gazillion widgets to evaluate pick random samples, plot the means, and see if they are normally distributed.

In reliability and manufacturing I never saw an exception. Once you eliminate or reduce systemic problems, human and manufacturing processes tend to be normal. At least close enough to apply normal statistics.

Way back I used random number tables to derive a sample sequence. People used things like closing stock market numbers to derives a staring point in the tables.

There is a similar principle in reliability engineering. If a system is complex enough once the early failures are gone regardless of the underlying failure distributions of components the failure rate per unit of time will be approximately constant.

The 'bath tub curve'. Well demonstrated.
 
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