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'Gamma random variable'
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| Title of object: |
Gamma random variable |
| Canonical Name: |
GammaRandomVariable |
| Type: |
Definition |
| Created on: |
2001-10-26 04:28:48 |
| Modified on: |
2003-12-11 16:01:52 |
| Classification: |
msc:60-00, msc:62-00 |
| Synonyms: |
Gamma random variable=gamma distribution |
Revision comment (for changes between this and next version):
| Changes for correction #10331 ('capitalization'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
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\usepackage{xypic} |
Content:
A \textbf{Gamma random variable} with parameters $\alpha>0$ and $\lambda>0$ is one whose probability density function is given by
\begin{align*}
f_X(x) = \frac{ \lambda^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\lambda x}
\end{align*}
for $x>0$, and is denoted by $X\sim Gamma(\alpha, \lambda)$.
Notes:\\
\begin{enumerate}
\item 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.
\item The function $\Gamma: [0,\infty] \to R$ is the Gamma function, defined as $\Gamma(t) = \int_{0}^{\infty}{x^{t-1} e^{-x} dx}$.
\item The expected value of a Gamma random variable is given by $E[X]=\frac{\alpha}{\lambda}$, and the variance by $Var[X] = \frac{\alpha}{\lambda^2}$
\item The moment generating function of a Gamma random variable is given by $M_X(t) = (\frac{\lambda}{\lambda - t})^\alpha$.
\end{enumerate}
If the first parameter is a positive integer, the variate is usually called Erlang random variate. The sum of $n$ exponentially distributed variables with parameter $\lambda$ is a Gamma (Erlang) variate with parameters $n, \lambda$. |
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