|
|
|
Viewing Version
3
of
'chi-squared random variable'
|
[ view 'chi-squared random variable'
|
back to history
]
| Title of object: |
chi-squared random variable |
| Canonical Name: |
ChiSquaredRandomVariable |
| Type: |
Definition |
| Created on: |
2001-10-26 18:52:49 |
| Modified on: |
2004-07-08 17:28:56 |
| Classification: |
msc:60-00 |
| Synonyms: |
chi-squared random variable=central chi-squared distribution |
Revision comment (for changes between this and next version):
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
%\usepackage{graphicx}
%\usepackage{xypic} |
Content:
\PMlinkescapeword{degrees}
\PMlinkescapeword{intervals}
\PMlinkescapeword{represents}
\PMlinkescapeword{syntax}
$X$ is a \emph{central chi-squared random variable} with $n$ degrees of freedom if\\
\par
$f_X(x) =\frac{ (\frac{1}{2})^{\frac{n}{2}} } {\Gamma(\frac{n}{2})} x^{\frac{n}{2} - 1} e^{- \frac{1}{2} x} $, $x > 0$ \\
\par
Where $\Gamma$ represents the gamma function.
\par
Parameters:\\
\par
\begin{list}{$\star$ }{}
\item $n \in N$
\end{list}
\par
Syntax:\\
\par
$X\sim \chi_{(n)}^{2}$\\
\par
Notes:\\
\par
\begin{enumerate}
\item This distribution is very widely used in statistics, like in hypothesis tests and confidence intervals.
\item The chi-squared distribution with n degrees of freedom is a result of evaluating the gamma distribution with $\alpha = \frac{n}{2}$ and $\lambda = \frac{1}{2}$.
\item $E[X] = n$
\item $Var[X] = 2n$
\item $M_X(t) = ( \frac{1}{1 - 2t} )^{\frac{n}{2}}$
\end{enumerate} |
|
|
|
|
|
