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 .
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)
If
Z
1
,
…
,
Z
k
{\displaystyle Z_{1},\ldots ,Z_{k}}
k
{\displaystyle k}
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}
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).
Probability density function
edit
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)}
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)\,}
where
P
(
k
,
x
)
{\displaystyle P(k,x)}
regularized gamma function .
Generating functions
edit
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)}
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).}
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)\ }
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))}
where
ψ
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 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]}
If
X
∼
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. if
X
∼
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 if
Y
∼
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 if
Y
∼
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 a standard normal random vector of with
k
{\displaystyle k}
k
{\displaystyle k}
degrees of freedom chi distribution is a special case of the generalized gamma distribution or the Nakagami distribution or the noncentral chi distribution
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}
unbiased estimation of the standard deviation of the normal distribution .
Various chi and chi-squared distributions
Name
Statistic
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}}}}
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 . 
![{\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}}})]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75ab2d5763ab73955d1da591a2c76909c453fb48)
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f3e1534e28ac6e9276477260c929f0fa2072a4e)
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f137a4f4c36c6d1a3e6e133d9ea9c8cf4cc3cfa)
![{\displaystyle ={\sqrt {n-1}}\,\cdot \left[1-{\frac {1}{4n}}+O({\frac {1}{n^{2}}})\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16110f5e6cee819cf7ce548062a398bcf57f7eac)
![{\displaystyle V=(n-1)-\mu ^{2}\,=(n-1)\cdot {\frac {1}{2n}}\,\cdot \left[1+O({\frac {1}{n}})\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5610836c34d335011ebc02dc4ee6ec6753aac429)
