IfLamun <math>p</math> independentwatesan linearlinier homogeneoushomogen constraintsbebas areditumpukeun imposeddina onieu these variablesvariabel, the distributionkayaan ofsebaran <math>X</math> conditionaldina onwatesan theseieu constriants isnyaeta <math>\chi^2_{k-p}</math>, justifyingdipastikeun thesalaku termwatesan "degreestingkat of freedomkabebasan". The [[characteristicCharacteristic function]] of thesebaran Chi-square distributionkuadrat isnyaeta
:<math>
\phi(t)=(1-2it)^{k/2}.</math>
TheSebaran chi-square distributionkuadrat hasngabogaan numerousaplikasi applicationsnumeris indina inferentialkaputusan [[statistik]], forcontona instance indina [[chi-square test]]s andinjeung estimatingestimasi [[variance]]s. ItIeu entersbisa thediasupkeun problemkana ofmasalah estimating theestimasi mean ofdina apopulasi normallysebaran distributednormal populationjeung andmasalah the problem of estimating theestimasi slope ofdina agaris [[linear regression|regression]] lineku viaaturan its role indina [[Student'ssebaran-t t-distributionstudent]]. ItIeu diasupkeun kana enterssakabeh allmasalah [[analysis of variance]] problemsku viaaturan its role in thedina [[sebaran F-distribution]], whichnu isngarupakeun thesebaran distributionperbandingan of the ratio of twodua chi-squaredkuadrat [[variabel random variable]]s.