Chi distribution

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]| cdf =[P(k/2,x^2/2)\,]| mean =[\mu=\sqrt\,\frac]| median =| mode =[\sqrt\,] for [k\ge 1]| variance =[\sigma^2=k-\mu^2\,]| skewness =[\gamma_1=\frac\,(1-2\sigma^2)]| kurtosis =[\frac(1-\mu\sigma\gamma_1-\sigma^2)]| entropy =[\ln(\Gamma(k/2))+\,][\frac(k\!-\!\ln(2)\!-\!(k\!-\!1)\psi_0(k/2))]| mgf =Complicated (see text)| char =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] are k independent, normally distributed random variables with means [\mu_i] and variances [\sigma_i], then the statistic

[Z = \sqrt\right)^2}]

is distributed according to the chi distribution. The chi distribution has one parameter: [k] which specifies the number of degrees of freedom (i.e. the number of [X_i]).

Properties

The probability density function is

[f(x;k) = \fracx^e^}]

where [\Gamma(z)] is the Gamma function. The cumulative distribution function is given by:

[F(x;k)=P(k/2,x^2/2)\,]

where [P(k,x)] is the regularized Gamma function. The moment generating function is given by:

[M(t)=M\left(\frac,\frac,\frac\right)+]

[t\sqrt\,\fracM\left(\frac,\frac,\frac\right)]

where [M(a,b,z)] is Kummer's confluent hypergeometric function. The raw moments are then given by:

[\mu_j=2^\frac]

where [\Gamma(z)] is the Gamma function. The first few raw moments are:

[\mu_1=\sqrt\,\,\frac]

[\mu_2=k\,]

[\mu_3=2\sqrt\,\,\frac=(k+1)\mu_1]

[\mu_4=(k)(k+2)\,]

[\mu_5=4\sqrt\,\,\frac=(k+1)(k+3)\mu_1]

[\mu_6=(k)(k+2)(k+4)\,]

where the rightmost expressions are derived using the recurrence relationship for the Gamma function:

[\Gamma(x+1)=x\Gamma(x)\,]

From these expressions we may derive the following relationships:

Mean: [\mu=\sqrt\,\,\frac]

Variance: [\sigma^2=k-\mu^2\,]

Skewness: [\gamma_1=\frac\,(1-2\sigma^2)]

Kurtosis excess: [\gamma_2=\frac(1-\mu\sigma\gamma_1-\sigma^2)]

The characteristic function is given by:

[\varphi(t;k)=M\left(\frac,\frac,\frac\right)+]

[it\sqrt\,\fracM\left(\frac,\frac,\frac\right)]

where again, [M(a,b,z)] is Kummer's confluent hypergeometric function. The entropy is given by:

[S=\ln(\Gamma(k/2))+\frac(k\!-\!\ln(2)\!-\!(k\!-\!1)\psi_0(k/2))]

where [\psi_0(z)] is the Polygamma function.

Related distributions

• If [X] is chi distributed [X \sim \chi_k(x)] then [X^2] is chi-square distributed: [X^2 \sim \chi^2_k]
• The Rayleigh distribution with [\sigma=1] is a chi distribution with two degrees of freedom.
• The Maxwell distribution for normalized molecular speeds is a chi distribution with three degrees of freedom.
• The chi distribution for [k=1] is the half-normal distribution.

Various chi and chi-square distributions

Name	Statistic
chi-square distribution	[\sum_1^k \left(\frac\right)^2]
noncentral chi-square distribution	[\sum_1^k \left(\frac\right)^2]
chi distribution	[\sqrt\right)^2}]
noncentral chi distribution	[\sqrt\right)^2}]

 

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