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The sum of n geometric random variables with probability of success p is a negative binomial random variable with parameters n and p. The sum of n exponential (β) random variables is a gamma (n, β) random variable. Since n is an integer, the gamma distribution is also a Erlang distribution.
Indeed, we know that if X is an exponential r.v. with rate λ, then cX is an exponential r.v. with rate λ/c; the same thing is valid with Gamma variates (and this can be checked using the moment-generating function, see, e.g.,these notes, 10.4-(ii)): multiplication by a positive constant c divides the rate (or, equivalently, multiplies the scale).
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 ...
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]
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
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
Exponential functions with bases 2 and 1/2. In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative equal to its value. . The exponential of a variable is denoted or , with the two notations used interchangeab