Probability distribution
chi
Probability density function
Cumulative distribution function
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. It is thus related to the
chi-squared distribution by describing the distribution of 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
Probability density function
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
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
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
Moments
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
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
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
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 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}
) yields the correction factor in the
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}}}}
See also
References
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
Discrete univariate
with finite support with infinite support
Continuous univariate
supported on a bounded interval supported on a semi-infinite interval supported on the whole real line with support whose type varies
Mixed univariate
Multivariate (joint)
Directional
Degenerate and
singular Families