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  2. Central limit theorem - Wikipedia

    en.wikipedia.org/wiki/Central_limit_theorem

    In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard normal distribution. This holds even if the original variables themselves are not normally distributed. There are several versions of the CLT, each applying in the ...

  3. Z-test - Wikipedia

    en.wikipedia.org/wiki/Z-test

    Because of the central limit theorem, many test statistics are approximately normally distributed for large samples.Therefore, many statistical tests can be conveniently performed as approximate Z-tests if the sample size is large or the population variance is known.

  4. Characteristic function (probability theory) - Wikipedia

    en.wikipedia.org/wiki/Characteristic_function...

    The characteristic function approach is particularly useful in analysis of linear combinations of independent random variables: a classical proof of the Central Limit Theorem uses characteristic functions and Lévy's continuity theorem. Another important application is to the theory of the decomposability of random variables.

  5. Normal distribution - Wikipedia

    en.wikipedia.org/wiki/Normal_distribution

    Comparison of probability density functions, () for the sum of fair 6-sided dice to show their convergence to a normal distribution with increasing , in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).

  6. Independent and identically distributed random variables

    en.wikipedia.org/wiki/Independent_and...

    The i.i.d. assumption is also used in the central limit theorem, which states that the probability distribution of the sum (or average) of i.i.d. variables with finite variance approaches a normal distribution. [4] The i.i.d. assumption frequently arises in the context of sequences of random variables. Then, "independent and identically ...

  7. Asymptotic distribution - Wikipedia

    en.wikipedia.org/wiki/Asymptotic_distribution

    The central limit theorem gives only an asymptotic distribution. As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.

  8. Illustration of the central limit theorem - Wikipedia

    en.wikipedia.org/wiki/Illustration_of_the...

    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

  9. Stein's method - Wikipedia

    en.wikipedia.org/wiki/Stein's_method

    Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric.It was introduced by Charles Stein, who first published it in 1972, [1] to obtain a bound between the distribution of a sum of -dependent sequence of random variables and a standard normal distribution in the Kolmogorov (uniform ...