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In probability theory and statistics, the χ 2 {\displaystyle \chi ^{2}} -distribution with k {\displaystyle k} degrees of freedom is the distribution of a sum of the squares of k {\displaystyle k} independent standard normal random variables.
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distribution displaystyle chi-squared chi normal sim frac gamma random independent textstyle variables freedom sum standard function degrees mean probability distributed
| Subject | Predicate | Object | Confidence | Src |
|---|---|---|---|---|
| Chi-squared distribution | CDF | 1 Γ ( k / 2 ) γ ( k 2 , x 2 ) {\displaystyle {\frac {1}{\Gamma (k/2)}}\;\gamma {\left({\frac {k}{2}},\,{\frac {x}{2}}\right)}\;} | 1.00 | infobox |
| Chi-squared distribution | CF | ( 1 − 2 i t ) − k / 2 {\displaystyle (1-2it)^{-k/2}} | 1.00 | infobox |
| Chi-squared distribution | Entropy | k 2 + log ( 2 Γ ( k 2 ) ) + ( 1 − k 2 ) ψ ( k 2 ) {\displaystyle {\begin{aligned}{\frac {k}{2}}&+\log \left(2\Gamma {\left({\frac {k}{2}}\right)}\right)\\&\!+\left(1-{\frac {k… | 1.00 | infobox |
| Chi-squared distribution | Excess kurtosis | 12 k {\displaystyle {\frac {12}{k}}} | 1.00 | infobox |
| Chi-squared distribution | Mean | k {\displaystyle k} | 1.00 | infobox |
| Chi-squared distribution | Median | ≈ k ( 1 − 2 9 k ) 3 {\displaystyle \approx k{\bigg (}1-{\frac {2}{9k}}{\bigg )}^{3}\;} | 1.00 | infobox |
| Chi-squared distribution | MGF | ( 1 − 2 t ) − k / 2 {\displaystyle (1-2t)^{-k/2}} for t < 1 2 {\displaystyle t<{\tfrac {1}{2}}\;} | 1.00 | infobox |
| Chi-squared distribution | Mode | max ( k − 2 , 0 ) {\displaystyle \max(k-2,0)\;} | 1.00 | infobox |
| Chi-squared distribution | Notation | χ 2 ( k ) {\displaystyle \chi ^{2}(k)\;} or χ k 2 {\displaystyle \chi _{k}^{2}\!} | 1.00 | infobox |
| Chi-squared distribution | Parameters | k ∈ N ∗ {\displaystyle k\in \mathbb {N} ^{*}~~} (known as "degrees of freedom") | 1.00 | infobox |
| Chi-squared distribution | 1 2 k / 2 Γ ( k / 2 ) x ( k / 2 ) − 1 e − x / 2 {\displaystyle {\frac {1}{2^{k/2}\Gamma (k/2)}}\;x^{(k/2)-1}e^{-x/2}\;} | 1.00 | infobox | |
| Chi-squared distribution | PGF | ( 1 − 2 ln t ) − k / 2 {\displaystyle (1-2\ln t)^{-k/2}} for 0 < t < e {\displaystyle 0<t<{\sqrt {e}}\;} | 1.00 | infobox |
| Chi-squared distribution | Skewness | 8 / k {\textstyle {\sqrt {8/k}}\,} | 1.00 | infobox |
| Chi-squared distribution | Support | x ∈ ( 0 , + ∞ ) {\displaystyle x\in (0,+\infty )\;} | 1.00 | infobox |
| Chi-squared distribution | Variance | 2 k {\displaystyle 2k\;} | 1.00 | infobox |
| Chi-squared distribution | is a | square of a standard normal distribution | 0.90 | text |
| Chi-squared distribution | is a | maximum entropy probability distribution for a random variate X | 0.90 | text |
| Chi-squared distribution | is a | sum of k | 0.90 | text |
| Chi-squared distribution | is a | special case of type III Pearson distributionIf X | 0.90 | text |
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