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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 ...
A graphical tool for assessing normality is the normal probability plot, a quantile-quantile plot (QQ plot) of the standardized data against the standard normal distribution. Here the correlation between the sample data and normal quantiles (a measure of the goodness of fit) measures how well the data are modeled by a normal distribution. For ...
A special case is the discrete distribution of a random variable that can take on only one fixed value; in other words, it is a deterministic distribution. Expressed formally, the random variable X {\displaystyle X} has a one-point distribution if it has a possible outcome x {\displaystyle x} such that P ( X = x ) = 1. {\displaystyle P(X{=}x)=1 ...
By contrast, the normal distribution, being a continuous distribution, has no discrete part—that is, it does not concentrate more than zero probability at any single point. Consequently X {\displaystyle X} and Y {\displaystyle Y} are not jointly normally distributed, even though they are separately normally distributed.
The Bates distribution is the distribution of the mean of n independent random variables, each of which having the uniform distribution on [0,1]. The logit-normal distribution on (0,1). The Dirac delta function, although not strictly a
If X 1 is a normal (μ 1, σ 2 1) random variable and X 2 is a normal (μ 2, σ 2 2) random variable, then X 1 + X 2 is a normal (μ 1 + μ 2, σ 2 1 + σ 2 2) random variable. The sum of N chi-squared (1) random variables has a chi-squared distribution with N degrees of freedom. Other distributions are not closed under convolution, but their ...
Clearly, the normal distribution is stable, but there are also other stable distributions, such as the Cauchy distribution, for which the mean or variance are not defined. The scaling factor b n may be proportional to n c , for any c ≥ 1 / 2 ; it may also be multiplied by a slowly varying function of n .
The "central" versions are derived from normal distributions that have a mean of zero; the noncentral versions generalize to arbitrary means. For example, the standard (central) chi-squared distribution is the distribution of a sum of squared independent standard normal distributions, i.e., normal distributions with mean 0, variance 1.