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More generally, if X 1 is a gamma(α 1, β 1) random variable and X 2 is an independent gamma(α 2, β 2) random variable then β 2 X 1 /(β 2 X 1 + β 1 X 2) is a beta(α 1, α 2) random variable. If X and Y are independent exponential random variables with mean μ, then X − Y is a double exponential random variable with mean 0 and scale μ.
The sum of exponentials is a useful model in pharmacokinetics (chemical kinetics in general) for describing the concentration of a substance over time. The exponential terms correspond to first-order reactions, which in pharmacology corresponds to the number of modelled diffusion compartments. [2] [3]
Suppose we wish to generate random variables from Gamma(n + δ, 1), where n is a non-negative integer and 0 < δ < 1. Using the fact that a Gamma(1, 1) distribution is the same as an Exp(1) distribution, and noting the method of generating exponential variables, we conclude that if U is uniformly distributed on (0, 1], then −ln U is ...
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
In probability theory, an exponentially modified Gaussian distribution (EMG, also known as exGaussian distribution) describes the sum of independent normal and exponential random variables. An exGaussian random variable Z may be expressed as Z = X + Y, where X and Y are independent, X is Gaussian with mean μ and variance σ 2, and Y is ...
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here, is taken to have the value
For example, the Bernoulli distribution is a binomial distribution with n = 1 trial, the exponential distribution is a gamma distribution with shape parameter α = 1 (or k = 1 ), and the geometric distribution is a special case of the negative binomial distribution. Some exponential family distributions are not NEF.
In the general case where there are distinct sums of exponential distributions with rates ,,, and a number of terms in each sum equals to ,,, respectively. The cumulative distribution function for t ≥ 0 {\displaystyle t\geq 0} is given by