Normal Distribution

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Normal Distribution

The well known bell shaped curve. According to the Central Limit Theorem, the probability density function of a large number of independent, identically distributed random numbers will approach the normal distribution. In the fractal family of distributions, the normal distribution only exists when alpha equals 2, or the Hurst exponent equals 0.50. Thus, the normal distribution is a special case which in time series analysis is quite rare. See: Alpha, Central Limit Theorem, Fractal Distribution.

Bell Curve

A curve on a chart in which most data points cluster around the median and become less frequent the farther they fall to either side of the median. When plotted on a chart, a bell curve looks roughly like a bell.
References in periodicals archive ?
(4) Based on the lower triangular matrix L, the inde pendent normal random variables Y = ([Y.sub.1], [Y.sub.2], ..., [Y.sub.n]) were transformed into standard normal distribution variables Z = ([Z.sub.1], [Z.sub.2], ..., [Z.sub.n])
The Y axis of cumulative standard normal distribution is divided into two parts by Moro algorithm, and then takes two corresponding algorithms for processing.
where [phi](t) denotes the pdf of the standard normal distribution.
From the standard normal distribution table we get the value Z = 1.765 matching the probability value 0.0389.
where [phi] is the standard normal distribution function and [phi] is the cumulative standard normal distribution function.
where [Z.sub.1-[alpha]/2] is the (1-[alpha]/2)th quantile of the standard normal distribution.
Agresti introduced a score [v.sub.j], and let [v.sub.j] = [[PHI].sup.- 1]([r.sub.j]), where [PHI] is a cumulative distribution function for standard normal distribution and [r.sub.j] is the Ridit score in category j [4].
where [PHI](x) represents the cumulative distribution function of the standard normal distribution.
In general, the density function of STN distribution can be represented as 2[[omega].sup.-1]t(z; v)[PHI]([alpha]z), where t and [PHI], respectively, denote the univariate standard Students t density function and the univariate standard normal distribution function and v is the degrees of freedom.
where [z.sub.[alpha]/2] is the upper 100([alpha]/2) percentile of the standard normal distribution and e = (1 - [[??].sup.2] + [omega]/2)/(1 - [[??].sup.2] - [omega]/2).
Moreover, the normalized random variable [[N.sub.n] - E([N.sub.n])]/[square root of V([N.sub.n])] converges in law to a standard normal distribution.

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