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chi-squared random variable (Definition)

$ X$ is a central chi-squared random variable with $ n$ degrees of freedom if

$\displaystyle f_X(x) = \frac{ (\frac{1}{2})^{\frac{n}{2}} } {\Gamma(\frac{n}{2})} x^{\frac{n}{2} - 1} e^{- \frac{1}{2} x} $
for $ x > 0$, where $ \Gamma$ represents the gamma function.

Parameters: $ n \in \mathbb{N}$

Syntax: $ X\sim \chi_{(n)}^{2}$

Notes:

  1. This distribution is very widely used in statistics, like in hypothesis tests and confidence intervals.
  2. 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}$.
  3. $ E[X] = n$
  4. $ \operatorname{Var}[X] = 2n$
  5. $ M_X(t) = ( \frac{1}{1 - 2t} )^{\frac{n}{2}}$



"chi-squared random variable" is owned by mathcam. [ full author list (3) | owner history (1) ]
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See Also: chi-squared statistic

Other names:  central chi-squared distribution

Attachments:
non-central chi-squared random variable (Definition) by CWoo
table of critical values of chi-squared distributions (Data Structure) by CWoo
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Cross-references: gamma distribution, hypothesis, statistics, distribution, parameters, gamma function, degrees of freedom
There are 11 references to this entry.

This is version 5 of chi-squared random variable, born on 2001-10-26, modified 2006-09-27.
Object id is 551, canonical name is ChiSquaredRandomVariable.
Accessed 9109 times total.

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