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Since probability tables cannot be printed for every normal distribution, as there are an infinite variety of normal distributions, it is common practice to convert a normal to a standard normal (known as a z-score) and then use the standard normal table to find probabilities. [2]
The simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution. This is a special case when μ = 0 {\textstyle \mu =0} and σ 2 = 1 {\textstyle \sigma ^{2}=1} , and it is described by this probability density function (or density): φ ( z ) = e − z 2 2 2 π . {\displaystyle \varphi (z ...
The characteristic function of the normal distribution with expected value μ and variance σ 2 is φ ( t ) = exp ( i t μ − σ 2 t 2 2 ) . {\displaystyle \varphi (t)=\exp \left(it\mu -{\sigma ^{2}t^{2} \over 2}\right).}
Diagram showing the cumulative distribution function for the normal distribution with mean (μ) 0 and variance (σ 2) 1. These numerical values "68%, 95%, 99.7%" come from the cumulative distribution function of the normal distribution. The prediction interval for any standard score z corresponds numerically to (1 − (1 − Φ μ,σ 2 (z)) · 2).
Let z = (z 1, ..., z N) T be a vector whose components are N independent standard normal variates (which can be generated, for example, by using the Box–Muller transform). Let x be μ + Az . This has the desired distribution due to the affine transformation property.
There is no single accepted name for this number; it is also commonly referred to as the "standard normal deviate", "normal score" or "Z score" for the 97.5 percentile point, the .975 point, or just its approximate value, 1.96. If X has a standard normal distribution, i.e. X ~ N(0,1),
Looking up the z-score in a table of the standard normal distribution cumulative probability, we find that the probability of observing a standard normal value below −2.47 is approximately 0.5 − 0.4932 = 0.0068.
The distribution of the product of correlated non-central normal samples was derived by Cui et al. [11] and takes the form of an infinite series of modified Bessel functions of the first kind. Moments of product of correlated central normal samples. For a central normal distribution N(0,1) the moments are