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In probability theory and statistics, an inverse distribution is the distribution of the reciprocal of a random variable. Inverse distributions arise in particular in the Bayesian context of prior distributions and posterior distributions for scale parameters .
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.
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 .
Inverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, or the Smirnov transform) is a basic method for pseudo-random number sampling, i.e., for generating sample numbers at random from any probability distribution given its cumulative distribution function.
Here F X is the cumulative distribution function of X, f X is the corresponding probability density function, Q X (p) is the corresponding inverse cumulative distribution function also called the quantile function, [2] and the integrals are of the Riemann–Stieltjes kind.
If X has cumulative distribution function F X, then the inverse of the cumulative distribution F X (X) is a standard uniform (0,1) random variable; If X is a normal (μ, σ 2) random variable then e X is a lognormal (μ, σ 2) random variable. Conversely, if X is a lognormal (μ, σ 2) random variable then log X is a normal (μ, σ 2) random ...
The inverse Gaussian distribution is a two-parameter exponential family with natural parameters −λ/(2μ 2) and −λ/2, and natural statistics X and 1/X.. For > fixed, it is also a single-parameter natural exponential family distribution [2] where the base distribution has density
The inverse cumulative distribution function (quantile function) of the logistic distribution is a generalization of the logit function. Its derivative is called the quantile density function. They are defined as follows: (;,) = + ().