Chi-Squared Distribution | chi square distribution pdf
![Chi-Squared Distribution](https://i.imgur.com/axBPWDg.jpg)
Ifhavenormal[1]independent[2]distributionswithmean[3]0andvariance[4]1,then(1)isdistributedaswithdegreesoffreedom[5].Thismakesadistributionagammadistribution[6]withand,whereisthenumberofdegreesoffreedom[7].Moregenerally,ifareindependentlydistributedaccordingtoadistributionwith,,...,degreesoffreedom[8],then(2)isdistributedaccordingtowithdegreesoffreedom[9].Theprobabilitydensityfunctionforthedistributionwithdegreesoffreedomisgivenby(3)for,whereisagammafunction[10].Thecumulativedistributionfunct...
![Chi-Squared Distribution](https://i.imgur.com/H8YKUrg.jpg)
If have normal[1] independent[2] distributions with mean[3] 0 and variance[4] 1, then
(1)
is distributed as with degrees of freedom[5]. This makes a distribution a gamma distribution[6] with and , where is the number of degrees of freedom[7].
More generally, if are independently distributed according to a distribution with , , ..., degrees of freedom[8], then
(2)
is distributed according to with degrees of freedom[9].
The probability density function for the distribution with degrees of freedom is given by
(3)
for , where is a gamma function[10]. The cumulative distribution function is then
where is an incomplete gamma function[11] and is a regularized gamma function[12].
The chi-squared distribution is implemented in the Wolfram Language[13] as ChiSquareDistribution[14][n].
For , is monotonic decreasing, but for , it has a maximum at