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Statistics:
Scientific method ·
Research methods ·
Experimental design ·
Undergraduate statistics courses ·
Statistical tests ·
Game theory ·
Decision theory
chi
Probability density function
Cumulative distribution function
Parameters
k
>
0
{\displaystyle k > 0\,}
Support
x
∈
[
0
;
∞
)
{\displaystyle x\in [0;\infty)}
pdf
2
1
−
k
/
2
x
k
−
1
e
−
x
2
/
2
Γ
(
k
/
2
)
{\displaystyle \frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/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
Mode
k
−
1
{\displaystyle \sqrt{k-1}\,}
k
≥
1
{\displaystyle k \ge 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)}
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)
Char. func. Complicated (see text)
In probability theory and statistics , the chi distribution is a continuous probability distribution . The distribution usually arises when a k-dimensional vector's orthogonal components are independent and each follow a standard normal distribution. The length of the vector will then have a chi distribution. The most familiar example is the Maxwell distribution of (normalized) molecular speeds which is a chi distribution with 3 degrees of freedom. If
X
i
{\displaystyle X_i}
k independent, normally distributed random variables with means
μ
i
{\displaystyle \mu_i}
standard deviations
σ
i
{\displaystyle \sigma_i}
Z
=
∑
1
k
(
X
i
−
μ
i
σ
i
)
2
{\displaystyle Z = \sqrt{\sum_1^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}}
is distributed according to the chi distribution. The chi distribution has one parameter:
k
{\displaystyle k}
degrees of freedom (i.e. the number of
X
i
{\displaystyle X_i}
Properties [ ] The probability density function is
f
(
x
;
k
)
=
2
1
−
k
/
2
x
k
−
1
e
−
x
2
/
2
Γ
(
k
/
2
)
{\displaystyle f(x;k) = \frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}}
where
Γ
(
z
)
{\displaystyle \Gamma(z)}
Gamma 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)}
regularized Gamma function . The moment generating function is given by:
M
(
t
)
=
M
(
k
2
,
1
2
,
t
2
2
)
+
{\displaystyle M(t)=M\left(\frac{k}{2},\frac{1}{2},\frac{t^2}{2}\right)+}
t
2
Γ
(
(
k
+
1
)
/
2
)
Γ
(
k
/
2
)
M
(
k
+
1
2
,
3
2
,
t
2
2
)
{\displaystyle 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 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 characteristic function is given by:
φ
(
t
;
k
)
=
M
(
k
2
,
1
2
,
−
t
2
2
)
+
{\displaystyle \varphi(t;k)=M\left(\frac{k}{2},\frac{1}{2},\frac{-t^2}{2}\right)+}
i
t
2
Γ
(
(
k
+
1
)
/
2
)
Γ
(
k
/
2
)
M
(
k
+
1
2
,
3
2
,
−
t
2
2
)
{\displaystyle 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)}
where again,
M
(
a
,
b
,
z
)
{\displaystyle M(a,b,z)}
confluent hypergeometric function . 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 [ ] If
X
{\displaystyle X}
X
∼
χ
k
(
x
)
{\displaystyle X \sim \chi_k(x)}
X
2
{\displaystyle X^2}
chi-square distributed:
X
2
∼
χ
k
2
{\displaystyle X^2 \sim \chi^2_k}
The Rayleigh distribution with
σ
=
1
{\displaystyle \sigma = 1}
The Maxwell distribution for normalized molecular speeds is a chi distribution with three degrees of freedom.
The chi distribution for
k
=
1
{\displaystyle k = 1}
half-normal distribution.
Various chi and chi-square distributions
Name
Statistic
chi-square distribution
∑
1
k
(
X
i
−
μ
i
σ
i
)
2
{\displaystyle \sum_1^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}
noncentral chi-square distribution
∑
1
k
(
X
i
σ
i
)
2
{\displaystyle \sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2}
chi distribution
∑
1
k
(
X
i
−
μ
i
σ
i
)
2
{\displaystyle \sqrt{\sum_1^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}}
noncentral chi distribution
∑
1
k
(
X
i
σ
i
)
2
{\displaystyle \sqrt{\sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2}}
