In probability theory and statistics , the chi distribution is a continuous probability distribution . It is the distribution of the positive square root of the sum of squares of a set of independent random variables each following a standard normal distribution , or equivalently, the distribution of the Euclidean distance of the random variables from 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.
chi
Probability density function
Cumulative distribution function
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})}
Ex. 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. Accordingly, dividing by 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 . The chi distribution has one parameter,
k
{\displaystyle k}
degrees of freedom (i.e. the number of
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 Edit
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).}
Properties Edit
The raw moments are then given by:
μ
j
=
2
j
/
2
Γ
(
(
k
+
j
)
/
2
)
Γ
(
k
/
2
)
{\displaystyle \mu _{j}=2^{j/2}{\frac {\Gamma ((k+j)/2)}{\Gamma (k/2)}}}
where
Γ
(
z
)
{\displaystyle \Gamma (z)}
gamma function . The first few raw moments are:
μ
1
=
2
Γ
(
(
k
+
1
)
/
2
)
Γ
(
k
/
2
)
{\displaystyle \mu _{1}={\sqrt {2}}\,\,{\frac {\Gamma ((k\!+\!1)/2)}{\Gamma (k/2)}}}
μ
2
=
k
{\displaystyle \mu _{2}=k\,}
μ
3
=
2
2
Γ
(
(
k
+
3
)
/
2
)
Γ
(
k
/
2
)
=
(
k
+
1
)
μ
1
{\displaystyle \mu _{3}=2{\sqrt {2}}\,\,{\frac {\Gamma ((k\!+\!3)/2)}{\Gamma (k/2)}}=(k+1)\mu _{1}}
μ
4
=
(
k
)
(
k
+
2
)
{\displaystyle \mu _{4}=(k)(k+2)\,}
μ
5
=
4
2
Γ
(
(
k
+
5
)
/
2
)
Γ
(
k
/
2
)
=
(
k
+
1
)
(
k
+
3
)
μ
1
{\displaystyle \mu _{5}=4{\sqrt {2}}\,\,{\frac {\Gamma ((k\!+\!5)/2)}{\Gamma (k/2)}}=(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
(
(
k
+
1
)
/
2
)
Γ
(
k
/
2
)
{\displaystyle \mu ={\sqrt {2}}\,\,{\frac {\Gamma ((k+1)/2)}{\Gamma (k/2)}}}
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})}
Kurtosis excess:
γ
2
=
2
σ
2
(
1
−
μ
σ
γ
1
−
σ
2
)
{\displaystyle \gamma _{2}={\frac {2}{\sigma ^{2}}}(1-\mu \sigma \gamma _{1}-\sigma ^{2})}
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 .
Related distributions Edit
If
X
∼
χ
k
{\displaystyle X\sim \chi _{k}}
X
2
∼
χ
k
2
{\displaystyle X^{2}\sim \chi _{k}^{2}}
chi-squared 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 )If
X
∼
N
(
0
,
1
)
{\displaystyle X\sim N(0,1)\,}
|
X
|
∼
χ
1
{\displaystyle |X|\sim \chi _{1}\,}
If
X
∼
χ
1
{\displaystyle X\sim \chi _{1}\,}
σ
X
∼
H
N
(
σ
)
{\displaystyle \sigma X\sim HN(\sigma )\,}
half-normal distribution ) for any
σ
>
0
{\displaystyle \sigma >0\,}
χ
2
∼
R
a
y
l
e
i
g
h
(
1
)
{\displaystyle \chi _{2}\sim \mathrm {Rayleigh} (1)\,}
Rayleigh distribution )
χ
3
∼
M
a
x
w
e
l
l
(
1
)
{\displaystyle \chi _{3}\sim \mathrm {Maxwell} (1)\,}
Maxwell distribution )
‖
N
i
=
1
,
…
,
k
(
0
,
1
)
‖
2
∼
χ
k
{\displaystyle \|{\boldsymbol {N}}_{i=1,\ldots ,k}{(0,1)}\|_{2}\sim \chi _{k}}
2-norm of
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
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}}}}
References Edit
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 Edit 
