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Graph of the density of the inverse of the standard normal distribution. If variable X follows a standard normal distribution (,), then Y = 1/X follows a reciprocal standard normal distribution, heavy-tailed and bimodal, [2] with modes at and density
|parameters= — parameters of the distribution family (such as μ and σ 2 for the normal distribution). |support= — the support of the distribution, which may depend on the parameters. Specify this as <math>x \in some set</math> for continuous distributions, and as <math>k \in some set</math> for discrete distributions.
The normal distribution is perhaps the most important case. Because the normal distribution is a location-scale family, its quantile function for arbitrary parameters can be derived from a simple transformation of the quantile function of the standard normal distribution, known as the probit function. Unfortunately, this function has no closed ...
The class of normal-inverse Gaussian distributions is closed under convolution in the following sense: [9] if and are independent random variables that are NIG-distributed with the same values of the parameters and , but possibly different values of the location and scale parameters, , and ,, respectively, then + is NIG-distributed with parameters ,, + and +.
Plot of probit function. In probability theory and statistics, the probit function is the quantile function associated with the standard normal distribution.It has applications in data analysis and machine learning, in particular exploratory statistical graphics and specialized regression modeling of binary response variables.
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 probit function is denoted (), where () is the CDF of the standard normal distribution, as just mentioned: = /. As shown in the graph on the right, the logit and probit functions are extremely similar when the probit function is scaled, so that its slope at y = 0 matches the slope of the logit.
Its cumulant generating function (logarithm of the characteristic function) [contradictory] is the inverse of the cumulant generating function of a Gaussian random variable. To indicate that a random variable X is inverse Gaussian-distributed with mean μ and shape parameter λ we write X ∼ IG ( μ , λ ) {\displaystyle X\sim ...