Chi distribution

Probability distribution

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chi

Probability density function

Cumulative distribution function

Notation|  χ ( k ) {\displaystyle \chi (k)\;} or  χ k {\displaystyle \chi _{k}\\!}
Parameters|  k > 0 {\displaystyle k>0\,} (degrees of freedom)
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}}\,} for  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)

In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. Equivalently, it is the distribution of the Euclidean distance between a multivariate Gaussian random variable and the origin. The chi distribution describes the positive square roots of a variable obeying a chi-squared distribution.

If  Z 1 , … , Z k {\displaystyle Z_{1},\ldots ,Z_{k}} are  k {\displaystyle k} independent, normally distributed random variables with mean 0 and standard 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 parameter  k {\displaystyle k} , which specifies the degrees of freedom (i.e. the number of random variables  Z i {\displaystyle Z_{i}} ).

The most familiar examples are the Rayleigh distribution (chi distribution with two degrees of freedom) and the Maxwell–Boltzmann distribution of the molecular speeds in an ideal gas (chi distribution with three degrees of freedom).

## Definitions

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### Probability density function

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The probability 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}}}

where  Γ ( z ) {\displaystyle \Gamma (z)} is the gamma function.

### Cumulative distribution function

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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)\,}

where  P ( k , x ) {\displaystyle P(k,x)} is the regularized gamma function.

### Generating functions

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The moment-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),}

where  M ( a , b , z ) {\displaystyle M(a,b,z)} is Kummer's confluent hypergeometric function. The characteristic 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).}

## Properties

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### Moments

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The raw moments 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)\ } is the 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)}}\ ,} which is close to  k − 1 2 {\displaystyle {\sqrt {k-{\tfrac {1}{2}}\ }}\ } for large k.

Variance:  V = k − μ 2 , {\displaystyle V=k-\mu ^{2}\ ,} which approaches  1 2 {\displaystyle \ {\tfrac {1}{2}}\ } as 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)~.}

### Entropy

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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))}

where  ψ 0 ( z ) {\displaystyle \psi ^{0}(z)} is the polygamma function.

### Large n approximation

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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 the Legendre 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)}}}

Using Stirling'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

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* If  X ∼ χ k {\displaystyle X\sim \chi _{k}} then  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. if  X ∼ N ( 0 , 1 ) {\displaystyle X\sim N(0,1)\,} then  | X | ∼ χ 1 {\displaystyle |X|\sim \chi _{1}\,} , and if  Y ∼ H N ( σ ) {\displaystyle Y\sim \mathrm {HN} (\sigma )\,} for any  σ > 0 {\displaystyle \sigma >0\,} then  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 if  Y ∼ R a y l e i g h ( σ ) {\displaystyle Y\sim \mathrm {Rayleigh} (\sigma )\,} for any  σ > 0 {\displaystyle \sigma >0\,} then  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 if  Y ∼ M a x w e l l ( a ) {\displaystyle Y\sim \mathrm {Maxwell} (a)\,} for any  a > 0 {\displaystyle a>0\,} then  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}} , the Euclidean norm of a standard normal random vector of with  k {\displaystyle k} dimensions, is distributed according to a chi distribution with  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 of  n − 1 {\displaystyle n-1} ) yields the correction factor in the unbiased estimation of the standard deviation of the normal distribution.

Various chi and chi-squared distributions Name | Statistic

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chi-squared distribution |  ∑ i = 1 k ( X i − μ i σ i ) 2 {\displaystyle \sum _{i=1}^{k}\left({\frac {X_{i}-\mu _{i}}{\sigma _{i}}}\right)^{2}}
noncentral chi-squared distribution |  ∑ i = 1 k ( X i σ i ) 2 {\displaystyle \sum _{i=1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}
chi distribution |  ∑ i = 1 k ( X i − μ i σ i ) 2 {\displaystyle {\sqrt {\sum _{i=1}^{k}\left({\frac {X_{i}-\mu _{i}}{\sigma _{i}}}\right)^{2}}}}
noncentral chi distribution |  ∑ i = 1 k ( X i σ i ) 2 {\displaystyle {\sqrt {\sum _{i=1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}}}

## References

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* 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.

## External links

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* http://mathworld.wolfram.com/ChiDistribution.html

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Probability distributions (list)

Discrete
univariate|

| with finite
support|

* Benford
* Bernoulli
* Beta-binomial
* Binomial
* Categorical
* Hypergeometric
* Negative
* Poisson binomial
* Rademacher
* Soliton
* Discrete uniform
* Zipf
* Zipf–Mandelbrot

|

with infinite
support|

* Beta negative binomial
* Borel
* Conway–Maxwell–Poisson
* Discrete phase-type
* Delaporte
* Extended negative binomial
* Flory–Schulz
* Gauss–Kuzmin
* Geometric
* Logarithmic
* Mixed Poisson
* Negative binomial
* Panjer
* Parabolic fractal
* Poisson
* Skellam
* Yule–Simon
* Zeta

Continuous
univariate|

| supported on a
bounded interval|

* Arcsine
* ARGUS
* Balding–Nichols
* Bates
* Beta
* Generalized
* Beta rectangular
* Continuous Bernoulli
* Irwin–Hall
* Kumaraswamy
* Logit-normal
* Noncentral beta
* PERT
* Raised cosine
* Reciprocal
* Triangular
* U-quadratic
* Uniform
* Wigner semicircle

|

supported on a
semi-infinite
interval|

* Benini
* Benktander 1st kind
* Benktander 2nd kind
* Beta prime
* Burr
* Chi
* Chi-squared
* Noncentral
* Inverse
* Scaled
* Dagum
* Davis
* Erlang
* Hyper
* Exponential
* Hyperexponential
* Hypoexponential
* Logarithmic
* F
* Noncentral
* Folded normal
* Fréchet
* Gamma
* Generalized
* Inverse
* gamma/Gompertz
* Gompertz
* Shifted
* Half-logistic
* Half-normal
* Hotelling's T-squared
* Hartman–Watson
* Inverse Gaussian
* Generalized
* Kolmogorov
* Lévy
* Log-Cauchy
* Log-Laplace
* Log-logistic
* Log-normal
* Log-t
* Lomax
* Matrix-exponential
* Maxwell–Boltzmann
* Maxwell–Jüttner
* Mittag-Leffler
* Nakagami
* Pareto
* Phase-type
* Poly-Weibull
* Rayleigh
* Relativistic Breit–Wigner
* Rice
* Truncated normal
* type-2 Gumbel
* Weibull
* Discrete
* Wilks's lambda

supported
on the whole
real line|

* Cauchy
* Exponential power
* Fisher's z
* Kaniadakis κ-Gaussian
* Gaussian q
* Generalized hyperbolic
* Generalized logistic (logistic-beta)
* Generalized normal
* Geometric stable
* Gumbel
* Holtsmark
* Hyperbolic secant
* Johnson's SU
* Landau
* Laplace
* Asymmetric
* Logistic
* Noncentral t
* Normal (Gaussian)
* Normal-inverse Gaussian
* Skew normal
* Slash
* Stable
* Student's t
* Tracy–Widom
* Variance-gamma
* Voigt

with support
whose type varies|

* Generalized chi-squared
* Generalized extreme value
* Generalized Pareto
* Marchenko–Pastur
* Kaniadakis κ-exponential
* Kaniadakis κ-Gamma
* Kaniadakis κ-Weibull
* Kaniadakis κ-Logistic
* Kaniadakis κ-Erlang
* q-exponential
* q-Gaussian
* q-Weibull
* Shifted log-logistic
* Tukey lambda

Mixed
univariate|

| continuous-
discrete|

* Rectified Gaussian

|

Multivariate
(joint)|

* Discrete:
* Ewens
* Multinomial
* Dirichlet
* Negative
* Continuous:
* Dirichlet
* Generalized
* Multivariate Laplace
* Multivariate normal
* Multivariate stable
* Multivariate t
* Normal-gamma
* Inverse
* Matrix-valued:
* LKJ
* Matrix beta
* Matrix normal
* Matrix t
* Matrix gamma
* Inverse
* Wishart
* Normal
* Inverse
* Normal-inverse
* Complex
* Uniform distribution on a Stiefel manifold

Directional|

Univariate (circular) directional
Circular uniform
Univariate von Mises
Wrapped normal
Wrapped Cauchy
Wrapped exponential
Wrapped asymmetric Laplace
Wrapped Lévy
Bivariate (spherical)
Kent
Bivariate (toroidal)
Bivariate von Mises
Multivariate
von Mises–Fisher
Bingham

Degenerate
and singular|

Degenerate
Dirac delta function
Singular
Cantor

Families|

* Circular
* Compound Poisson
* Elliptical
* Exponential
* Natural exponential
* Location–scale
* Maximum entropy
* Mixture
* Pearson
* Tweedie
* Wrapped

* Category
* Commons

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