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In the shape-scale parametrization, X ~ Gamma(1, λ) has an exponential distribution with rate parameter 1/λ. If X ~ Gamma(ν/2, 2) (in the shape–scale parametrization), then X is identical to χ 2 (ν), the chi-squared distribution with ν degrees of freedom. Conversely, if Q ~ χ 2 (ν) and c is a positive constant, then cQ ~ Gamma(ν/2, 2c).
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]
The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic except at zero and the negative integers, where it has simple poles. The gamma function has no zeros, so the reciprocal gamma function 1 / Γ(z) is an entire function.
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 ...
2.2 Exponential function. ... 7.2 Sum of reciprocal of factorials. 7.3 Trigonometry and ... is the gamma function. is a polygamma ...
The area of the selection within the unit square and below the line z = xy, represents the CDF of z. This divides into two parts. The first is for 0 < x < z where the increment of area in the vertical slot is just equal to dx. The second part lies below the xy line, has y-height z/x, and incremental area dx z/x.
The gamma function is an important special function in mathematics.Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general.