Probability distribution
chi Probability density function
Cumulative distribution function
Notation χ ( k ) {\displaystyle \chi (k)\;} χ k {\displaystyle \chi _{k}\!} Parameters k > 0 {\displaystyle k>0\,} Support x ∈ [ 0 , ∞ ) {\displaystyle x\in [0,\infty )} PDF 1 2 ( k / 2 ) − 1 Γ ( k / 2 ) x k − 1 e − x 2 / 2 {\displaystyle {\frac {1}{2^{(k/2)-1}\Gamma (k/2)}}\;x^{k-1}e^{-x^{2}/2}} CDF P ( k / 2 , x 2 / 2 ) {\displaystyle P(k/2,x^{2}/2)\,} Mean μ = 2 Γ ( ( k + 1 ) / 2 ) Γ ( k / 2 ) {\displaystyle \mu ={\sqrt {2}}\,{\frac {\Gamma ((k+1)/2)}{\Gamma (k/2)}}} Median ≈ k ( 1 − 2 9 k ) 3 {\displaystyle \approx {\sqrt {k{\bigg (}1-{\frac {2}{9k}}{\bigg )}^{3}}}} Mode k − 1 {\displaystyle {\sqrt {k-1}}\,} k ≥ 1 {\displaystyle k\geq 1} Variance σ 2 = k − μ 2 {\displaystyle \sigma ^{2}=k-\mu ^{2}\,} Skewness γ 1 = μ σ 3 ( 1 − 2 σ 2 ) {\displaystyle \gamma _{1}={\frac {\mu }{\sigma ^{3}}}\,(1-2\sigma ^{2})} Excess kurtosis 2 σ 2 ( 1 − μ σ γ 1 − σ 2 ) {\displaystyle {\frac {2}{\sigma ^{2}}}(1-\mu \sigma \gamma _{1}-\sigma ^{2})} Entropy ln ( Γ ( k / 2 ) ) + {\displaystyle \ln(\Gamma (k/2))+\,} 1 2 ( k − ln ( 2 ) − ( k − 1 ) ψ 0 ( k / 2 ) ) {\displaystyle {\frac {1}{2}}(k\!-\!\ln(2)\!-\!(k\!-\!1)\psi _{0}(k/2))} MGF Complicated (see text) CF Complicated (see text)
Inprobability theory andstatistics , thechi distribution is a continuousprobability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independentGaussian random variables . Equivalently, it is the distribution of theEuclidean distance between a multivariate Gaussian random variable and the origin. The chi distribution describes the positive square roots of a variable obeying achi-squared distribution .
IfZ 1 , … , Z k {\displaystyle Z_{1},\ldots ,Z_{k}} k {\displaystyle k} normally distributed random variables with mean 0 andstandard deviation 1, then the statistic
Y = ∑ i = 1 k Z i 2 {\displaystyle Y={\sqrt {\sum _{i=1}^{k}Z_{i}^{2}}}} is distributed according to the chi distribution. The chi distribution has one positive integer parameterk {\displaystyle k} degrees of freedom (i.e. the number of random variablesZ i {\displaystyle Z_{i}}
The most familiar examples are theRayleigh distribution (chi distribution with twodegrees of freedom ) and theMaxwell–Boltzmann distribution of the molecular speeds in anideal gas (chi distribution with three degrees of freedom).
Probability density function [ edit ] Theprobability density function (pdf) of the chi-distribution is
f ( x ; k ) = { x k − 1 e − x 2 / 2 2 k / 2 − 1 Γ ( k 2 ) , x ≥ 0 ; 0 , otherwise . {\displaystyle f(x;k)={\begin{cases}{\dfrac {x^{k-1}e^{-x^{2}/2}}{2^{k/2-1}\Gamma \left({\frac {k}{2}}\right)}},&x\geq 0;\\0,&{\text{otherwise}}.\end{cases}}} whereΓ ( z ) {\displaystyle \Gamma (z)} gamma function .
Cumulative distribution function [ edit ] The cumulative distribution function is given by:
F ( x ; k ) = P ( k / 2 , x 2 / 2 ) {\displaystyle F(x;k)=P(k/2,x^{2}/2)\,} whereP ( k , x ) {\displaystyle P(k,x)} regularized gamma function .
Generating functions [ edit ] Themoment-generating function is given by:
M ( t ) = M ( k 2 , 1 2 , t 2 2 ) + t 2 Γ ( ( k + 1 ) / 2 ) Γ ( k / 2 ) M ( k + 1 2 , 3 2 , t 2 2 ) , {\displaystyle M(t)=M\left({\frac {k}{2}},{\frac {1}{2}},{\frac {t^{2}}{2}}\right)+t{\sqrt {2}}\,{\frac {\Gamma ((k+1)/2)}{\Gamma (k/2)}}M\left({\frac {k+1}{2}},{\frac {3}{2}},{\frac {t^{2}}{2}}\right),} whereM ( a , b , z ) {\displaystyle M(a,b,z)} confluent hypergeometric function . Thecharacteristic function is given by:
φ ( t ; k ) = M ( k 2 , 1 2 , − t 2 2 ) + i t 2 Γ ( ( k + 1 ) / 2 ) Γ ( k / 2 ) M ( k + 1 2 , 3 2 , − t 2 2 ) . {\displaystyle \varphi (t;k)=M\left({\frac {k}{2}},{\frac {1}{2}},{\frac {-t^{2}}{2}}\right)+it{\sqrt {2}}\,{\frac {\Gamma ((k+1)/2)}{\Gamma (k/2)}}M\left({\frac {k+1}{2}},{\frac {3}{2}},{\frac {-t^{2}}{2}}\right).} The rawmoments are then given by:
μ j = ∫ 0 ∞ f ( x ; k ) x j d x = 2 j / 2 Γ ( 1 2 ( k + j ) ) Γ ( 1 2 k ) {\displaystyle \mu _{j}=\int _{0}^{\infty }f(x;k)x^{j}\mathrm {d} x=2^{j/2}\ {\frac {\ \Gamma \left({\tfrac {1}{2}}(k+j)\right)\ }{\Gamma \left({\tfrac {1}{2}}k\right)}}} where Γ ( z ) {\displaystyle \ \Gamma (z)\ } gamma function . Thus the first few raw moments are:
μ 1 = 2 Γ ( 1 2 ( k + 1 ) ) Γ ( 1 2 k ) {\displaystyle \mu _{1}={\sqrt {2\ }}\ {\frac {\ \Gamma \left({\tfrac {1}{2}}(k+1)\right)\ }{\Gamma \left({\tfrac {1}{2}}k\right)}}} μ 2 = k , {\displaystyle \mu _{2}=k\ ,} μ 3 = 2 2 Γ ( 1 2 ( k + 3 ) ) Γ ( 1 2 k ) = ( k + 1 ) μ 1 , {\displaystyle \mu _{3}=2{\sqrt {2\ }}\ {\frac {\ \Gamma \left({\tfrac {1}{2}}(k+3)\right)\ }{\Gamma \left({\tfrac {1}{2}}k\right)}}=(k+1)\ \mu _{1}\ ,} μ 4 = ( k ) ( k + 2 ) , {\displaystyle \mu _{4}=(k)(k+2)\ ,} μ 5 = 4 2 Γ ( 1 2 ( k + 5 ) ) Γ ( 1 2 k ) = ( k + 1 ) ( k + 3 ) μ 1 , {\displaystyle \mu _{5}=4{\sqrt {2\ }}\ {\frac {\ \Gamma \left({\tfrac {1}{2}}(k\!+\!5)\right)\ }{\Gamma \left({\tfrac {1}{2}}k\right)}}=(k+1)(k+3)\ \mu _{1}\ ,} μ 6 = ( k ) ( k + 2 ) ( k + 4 ) , {\displaystyle \mu _{6}=(k)(k+2)(k+4)\ ,} where the rightmost expressions are derived using the recurrence relationship for the gamma function:
Γ ( x + 1 ) = x Γ ( x ) . {\displaystyle \Gamma (x+1)=x\ \Gamma (x)~.} From these expressions we may derive the following relationships:
Mean:μ = 2 Γ ( 1 2 ( k + 1 ) ) Γ ( 1 2 k ) , {\displaystyle \mu ={\sqrt {2\ }}\ {\frac {\ \Gamma \left({\tfrac {1}{2}}(k+1)\right)\ }{\Gamma \left({\tfrac {1}{2}}k\right)}}\ ,} k − 1 2 {\displaystyle {\sqrt {k-{\tfrac {1}{2}}\ }}\ } k .
Variance:V = k − μ 2 , {\displaystyle V=k-\mu ^{2}\ ,} 1 2 {\displaystyle \ {\tfrac {1}{2}}\ } k increases.
Skewness:γ 1 = μ σ 3 ( 1 − 2 σ 2 ) . {\displaystyle \gamma _{1}={\frac {\mu }{\ \sigma ^{3}\ }}\left(1-2\sigma ^{2}\right)~.}
Kurtosis excess:γ 2 = 2 σ 2 ( 1 − μ σ γ 1 − σ 2 ) . {\displaystyle \gamma _{2}={\frac {2}{\ \sigma ^{2}\ }}\left(1-\mu \ \sigma \ \gamma _{1}-\sigma ^{2}\right)~.}
The entropy is given by:
S = ln ( Γ ( k / 2 ) ) + 1 2 ( k − ln ( 2 ) − ( k − 1 ) ψ 0 ( k / 2 ) ) {\displaystyle S=\ln(\Gamma (k/2))+{\frac {1}{2}}(k\!-\!\ln(2)\!-\!(k\!-\!1)\psi ^{0}(k/2))} whereψ 0 ( z ) {\displaystyle \psi ^{0}(z)} polygamma function .
Large n approximation [ edit ] We find the large n=k+1 approximation of the mean and variance of chi distribution. This has application e.g. in finding the distribution of standard deviation of a sample of normally distributed population, where n is the sample size.
The mean is then:
μ = 2 Γ ( n / 2 ) Γ ( ( n − 1 ) / 2 ) {\displaystyle \mu ={\sqrt {2}}\,\,{\frac {\Gamma (n/2)}{\Gamma ((n-1)/2)}}} We use theLegendre duplication formula to write:
2 n − 2 Γ ( ( n − 1 ) / 2 ) ⋅ Γ ( n / 2 ) = π Γ ( n − 1 ) {\displaystyle 2^{n-2}\,\Gamma ((n-1)/2)\cdot \Gamma (n/2)={\sqrt {\pi }}\Gamma (n-1)} so that:
μ = 2 / π 2 n − 2 ( Γ ( n / 2 ) ) 2 Γ ( n − 1 ) {\displaystyle \mu ={\sqrt {2/\pi }}\,2^{n-2}\,{\frac {(\Gamma (n/2))^{2}}{\Gamma (n-1)}}} UsingStirling's approximation for Gamma function, we get the following expression for the mean:
μ = 2 / π 2 n − 2 ( 2 π ( n / 2 − 1 ) n / 2 − 1 + 1 / 2 e − ( n / 2 − 1 ) ⋅ [ 1 + 1 12 ( n / 2 − 1 ) + O ( 1 n 2 ) ] ) 2 2 π ( n − 2 ) n − 2 + 1 / 2 e − ( n − 2 ) ⋅ [ 1 + 1 12 ( n − 2 ) + O ( 1 n 2 ) ] {\displaystyle \mu ={\sqrt {2/\pi }}\,2^{n-2}\,{\frac {\left({\sqrt {2\pi }}(n/2-1)^{n/2-1+1/2}e^{-(n/2-1)}\cdot [1+{\frac {1}{12(n/2-1)}}+O({\frac {1}{n^{2}}})]\right)^{2}}{{\sqrt {2\pi }}(n-2)^{n-2+1/2}e^{-(n-2)}\cdot [1+{\frac {1}{12(n-2)}}+O({\frac {1}{n^{2}}})]}}} = ( n − 2 ) 1 / 2 ⋅ [ 1 + 1 4 n + O ( 1 n 2 ) ] = n − 1 ( 1 − 1 n − 1 ) 1 / 2 ⋅ [ 1 + 1 4 n + O ( 1 n 2 ) ] {\displaystyle =(n-2)^{1/2}\,\cdot \left[1+{\frac {1}{4n}}+O({\frac {1}{n^{2}}})\right]={\sqrt {n-1}}\,(1-{\frac {1}{n-1}})^{1/2}\cdot \left[1+{\frac {1}{4n}}+O({\frac {1}{n^{2}}})\right]} = n − 1 ⋅ [ 1 − 1 2 n + O ( 1 n 2 ) ] ⋅ [ 1 + 1 4 n + O ( 1 n 2 ) ] {\displaystyle ={\sqrt {n-1}}\,\cdot \left[1-{\frac {1}{2n}}+O({\frac {1}{n^{2}}})\right]\,\cdot \left[1+{\frac {1}{4n}}+O({\frac {1}{n^{2}}})\right]} = n − 1 ⋅ [ 1 − 1 4 n + O ( 1 n 2 ) ] {\displaystyle ={\sqrt {n-1}}\,\cdot \left[1-{\frac {1}{4n}}+O({\frac {1}{n^{2}}})\right]} And thus the variance is:
V = ( n − 1 ) − μ 2 = ( n − 1 ) ⋅ 1 2 n ⋅ [ 1 + O ( 1 n ) ] {\displaystyle V=(n-1)-\mu ^{2}\,=(n-1)\cdot {\frac {1}{2n}}\,\cdot \left[1+O({\frac {1}{n}})\right]} Related distributions [ edit ] IfX ∼ χ k {\displaystyle X\sim \chi _{k}} X 2 ∼ χ k 2 {\displaystyle X^{2}\sim \chi _{k}^{2}} chi-squared distribution ) χ 1 ∼ H N ( 1 ) {\displaystyle \chi _{1}\sim \mathrm {HN} (1)\,} half-normal distribution ), i.e. ifX ∼ N ( 0 , 1 ) {\displaystyle X\sim N(0,1)\,} | X | ∼ χ 1 {\displaystyle |X|\sim \chi _{1}\,} Y ∼ H N ( σ ) {\displaystyle Y\sim \mathrm {HN} (\sigma )\,} σ > 0 {\displaystyle \sigma >0\,} Y σ ∼ χ 1 {\displaystyle {\tfrac {Y}{\sigma }}\sim \chi _{1}\,} χ 2 ∼ R a y l e i g h ( 1 ) {\displaystyle \chi _{2}\sim \mathrm {Rayleigh} (1)\,} Rayleigh distribution ) and ifY ∼ R a y l e i g h ( σ ) {\displaystyle Y\sim \mathrm {Rayleigh} (\sigma )\,} σ > 0 {\displaystyle \sigma >0\,} Y σ ∼ χ 2 {\displaystyle {\tfrac {Y}{\sigma }}\sim \chi _{2}\,} χ 3 ∼ M a x w e l l ( 1 ) {\displaystyle \chi _{3}\sim \mathrm {Maxwell} (1)\,} Maxwell distribution ) and ifY ∼ M a x w e l l ( a ) {\displaystyle Y\sim \mathrm {Maxwell} (a)\,} a > 0 {\displaystyle a>0\,} Y a ∼ χ 3 {\displaystyle {\tfrac {Y}{a}}\sim \chi _{3}\,} ‖ N i = 1 , … , k ( 0 , 1 ) ‖ 2 ∼ χ k {\displaystyle \|{\boldsymbol {N}}_{i=1,\ldots ,k}{(0,1)}\|_{2}\sim \chi _{k}} Euclidean norm of astandard normal random vector of withk {\displaystyle k} k {\displaystyle k} degrees of freedom chi distribution is a special case of various distributions:generalized gamma ,Nakagami ,noncentral chi , etc. lim k → ∞ χ k − μ k σ k → d N ( 0 , 1 ) {\displaystyle \lim _{k\to \infty }{\tfrac {\chi _{k}-\mu _{k}}{\sigma _{k}}}{\xrightarrow {d}}\ N(0,1)\,} Normal distribution )The mean of the chi distribution (scaled by the square root ofn − 1 {\displaystyle n-1} unbiased estimation of the standard deviation of the normal distribution .
Martha L. Abell, James P. Braselton, John Arthur Rafter, John A. Rafter,Statistics with Mathematica (1999),237f. Jan W. Gooch,Encyclopedic Dictionary of Polymers vol. 1 (2010), Appendix E,p. 972 .
Discrete
with finite with infinite
Continuous
supported on a supported on a supported with support
Mixed
Multivariate Directional Degenerate singular Families
