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The inverse Gaussian distribution has several properties analogous to a Gaussian distribution. The name can be misleading: it is an "inverse" only in that, while the Gaussian describes a Brownian motion's level at a fixed time, the inverse Gaussian describes the distribution of the time a Brownian motion with positive drift takes to reach a ...
The quantile function, Q, of a probability distribution is the inverse of its cumulative distribution function F. The derivative of the quantile function, namely the quantile density function, is yet another way of prescribing a probability distribution. It is the reciprocal of the pdf composed with the quantile function.
The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing [41] (Matlab code). In the following sections we look at some special cases.
The Q-function can be generalized to higher dimensions: [14] = (),where (,) follows the multivariate normal distribution with covariance and the threshold is of the form = for some positive vector > and positive constant >.
Cumulative distribution function for the exponential distribution Cumulative distribution function for the normal distribution. In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable, or just distribution function of , evaluated at , is the probability that will take a value less than or equal to .
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 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 +.
The cumulative distribution function of the reciprocal, within the same range, is G ( y ) = b − y − 1 b − a . {\displaystyle G(y)={\frac {b-y^{-1}}{b-a}}.} For example, if X is uniformly distributed on the interval (0,1), then Y = 1 / X has density g ( y ) = y − 2 {\displaystyle g(y)=y^{-2}} and cumulative distribution function G ( y ...