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An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains the term "central limit theorem for convex bodies".
This section illustrates the central limit theorem via an example for which the computation can be done quickly by hand on paper, unlike the more computing-intensive example of the previous section. Sum of all permutations of length 1 selected from the set of integers 1, 2, 3
He named this his "Golden Theorem" but it became generally known as "Bernoulli's theorem". This should not be confused with Bernoulli's principle, named after Jacob Bernoulli's nephew Daniel Bernoulli. In 1837, S. D. Poisson further described it under the name "la loi des grands nombres" ("the law of large numbers").
The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example: The binomial distribution B ( n , p ) {\textstyle B(n,p)} is approximately normal with mean n p {\textstyle np} and variance n p ( 1 − p ) {\textstyle np(1-p)} for large n {\textstyle n} and for p {\textstyle p} not ...
An example of how is used is to ... provide a better bound on estimates of the population mean, in light of the central limit theorem. ... The formula given above for ...
2 Examples. 3 Properties. 4 Characterization. ... the quasi-arithmetic mean or generalised f-mean or Kolmogorov-Nagumo-de Finetti mean [1] ... Central limit theorem : ...
In probability theory, Lindeberg's condition is a sufficient condition (and under certain conditions also a necessary condition) for the central limit theorem (CLT) to hold for a sequence of independent random variables.
By definition, a consistent estimator B converges in probability to its true value β, and often a central limit theorem can be applied to obtain asymptotic normality: (,),