Between Certainty and Uncertainty Statistics and Probability in Five
„Between Certainty & Uncertainty” is a one-of–a-kind short course on statistics for students, engineers and researchers. It is a fascinating introduction to statistics and probability with notes on historical origins and 80 illustrative nu
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Normal Distribution Binomial Heritage
Acquaintance with the normal distribution, tables of the normal distribution. Probabilistic paper. Sample means distribution and Monte Carlo simulation. Two theorems of de Moivre-Laplace. When does normal approximation fit binomial distribution data? [Heritage of F.Gauss and Marquis de Laplace] –
5.1 Normal Statistics, Preliminaries To understand how specific and how universal the normal distribution is, the point based on the Central Limit Theorems of the Theory of Probability should be taken. A routine course presenting this theory usually closes with the Central Limit Theorems. Therefore, the foregoing presentation may only present these results without supplying the Student with any rigorous proofs and leaving out the details. Thus, when indicating possible courses which present a similar approach, let us first mention a book by Weinberg [1] referred to earlier rather than that by Neyman [2], however, this remark is addressed more to the instructor than to the student. From an intuitional point of view a very important element of the limit theorems seems to be the fact that the normal distribution is the result of the sum of a number of random components (strictly speaking they are random variables) not necessarily of precisely defined nature (which stands for the knowledge of their distributions)1. How universal the normal distribution is has constituted a heated subject of discussions or even bitter quarrels among mathematicians and statisticians for more than a century. There is no doubt about its power, but there is also no doubt about its limitations. The human species displays a wide range of such applications, from the purely physical (stature or weight) to mental (such as IQ or grades). One of its special applications is mass products. From a theoretical point of view the normal distribution has a unique property: invariance regarding linear transformations. The first encounter with such a property was offered by Chapter 1. 1
Just here we may recall a rule of the thumb well known in Statistics and using at least twelve uniformly distributed components to get a sample of the normal distribution.
L.M. Laudański: Between Certainty and Uncertainty, ISRL 31, pp. 129–157. springerlink.com © Springer-Verlag Berlin Heidelberg 2013
130
Normal Distribution Binomial Heritage
Below we commence with presenting the normal distribution from scratch. It is given in general form by (5.1) showing a real function of the real variable with
x , σ 2 and these symbols suggest the special meaning 2 of the parameters. Of course x denotes the basic mean, and σ denotes the
two parameters denoted by variance.
f ( x) =
1 e−( x− x ) σ 2π
2
/ 2σ 2
here:
−∞ < x < ∞
(5.1)
The above stated properties of the normal distribution will be proved (see p.5.4) – here we limit ourselves to presenting the defining steps. The basic mean is formally defined by:
μ=
+∞
x ⋅ f ( x) dx
(5.2)
−∞
To avoid a clash of symbols in (5.2) symbol μ also appears with respect to the first moment. The
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