PlanetMath: gamma random variable

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gamma random variable

(Definition)

A gamma random variable with parameters $\alpha>0$ and $\lambda>0$ is one whose probability density function is given by

$\displaystyle f_X(x) = \frac{ \lambda^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\lambda x}$

for

, and is denoted by $X\sim Gamma(\alpha, \lambda)$ .

Notes:

Gamma random variables are widely used in many applications. Taking $\alpha = 1$ reduces the form to that of an exponential random variable. If $\alpha = \frac{n}{2}$ and $\lambda = \frac{1}{2}$ , this is a chi-squared random variable.
The function $\Gamma: [0,\infty] \to R$ is the gamma function, defined as $\Gamma(t) = \int_{0}^{\infty}{x^{t-1} e^{-x} dx}$ .
The expected value of a gamma random variable is given by $E[X]=\frac{\alpha}{\lambda}$ , and the variance by $Var[X] = \frac{\alpha}{\lambda^2}$
The moment generating function of a gamma random variable is given by $M_X(t) = (\frac{\lambda}{\lambda - t})^\alpha$ .

If the first parameter is a positive integer, the variate is usually called Erlang random variate. The sum of exponentially distributed variables with parameter $\lambda$ is a gamma (Erlang) variate with parameters $n, \lambda$ .

"gamma random variable" is owned by mathcam. [ full author list (2) | owner history (1) ]

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Other names:

gamma distribution

Also defines:

Erlang random variable

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Cross-references: variables, sum, integer, positive, moment generating function, variance, expected value, gamma function, function, chi-squared random variable, exponential random variable, probability density function, parameters
There are 7 references to this entry.

This is version 9 of gamma random variable, born on 2001-10-26, modified 2006-10-25.
Object id is 529, canonical name is GammaRandomVariable.
Accessed 13530 times total.

Classification:

AMS MSC:	60-00 (Probability theory and stochastic processes :: General reference works )
	62-00 (Statistics :: General reference works )

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