Chi distribution
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In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, the chi distribution is a continuous
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
. 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 In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
, or equivalently, the distribution of the
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...
of the random variables from the origin. It is thus related to the
chi-squared distribution In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squa ...
by describing the distribution of the positive square roots of a variable obeying a chi-squared distribution. If Z_1, \ldots, Z_k are k independent, normally distributed random variables with mean 0 and
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
1, then the statistic :Y = \sqrt 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 random variables Z_i). The most familiar examples are the
Rayleigh distribution In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom. The distribut ...
(chi distribution with two degrees of freedom) and the
Maxwell–Boltzmann distribution In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and use ...
of the molecular speeds in an
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
(chi distribution with three degrees of freedom).


Definitions


Probability density function

The
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
(pdf) of the chi-distribution is :f(x;k) = \begin \dfrac, & x\geq 0; \\ 0, & \text. \end where \Gamma(z) is the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
.


Cumulative distribution 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 In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, whic ...
.


Generating functions

The
moment-generating function In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compare ...
is given by: :M(t)=M\left(\frac,\frac,\frac\right)+t\sqrt\,\frac M\left(\frac,\frac,\frac\right), where M(a,b,z) is Kummer's
confluent hypergeometric function In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular ...
. The
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at points ...
is given by: :\varphi(t;k)=M\left(\frac,\frac,\frac\right) + it\sqrt\,\frac M\left(\frac,\frac,\frac\right).


Properties


Moments

The raw moments are then given by: :\mu_j=\int_0^\infty f(x;k) x^j dx = 2^\frac where \Gamma(z) is the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
. Thus 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, which is close to \sqrt for large ''k'' Variance: V=k-\mu^2\,, which approaches \tfrac as ''k'' increases Skewness: \gamma_1=\frac\,(1-2\sigma^2) Kurtosis excess: \gamma_2=\frac(1-\mu\sigma\gamma_1-\sigma^2)


Entropy

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 In mathematics, the polygamma function of order is a meromorphic function on the complex numbers \mathbb defined as the th derivative of the logarithm of the gamma function: :\psi^(z) := \frac \psi(z) = \frac \ln\Gamma(z). Thus :\psi^(z) ...
.


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: :\mu = \sqrt\,\,\frac We use the Legendre duplication formula to write: :2^ \,\Gamma((n-1)/2)\cdot \Gamma(n/2) = \sqrt \Gamma (n-1), so that: :\mu = \sqrt\,2^\,\frac Using
Stirling's approximation In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less p ...
for Gamma function, we get the following expression for the mean: :\mu = \sqrt\,2^\,\frac :: = (n-2)^\,\cdot \left +\frac+O(\frac)\right= \sqrt\,(1-\frac)^\cdot \left +\frac+O(\frac)\right/math> :: = \sqrt\,\cdot \left -\frac+O(\frac)\right,\cdot \left +\frac+O(\frac)\right/math> :: = \sqrt\,\cdot \left -\frac+O(\frac)\right/math> And thus the variance is: :V=(n-1)-\mu^2\, = (n-1)\cdot \frac\,\cdot \left +O(\frac)\right/math>


Related distributions

*If X \sim \chi_k then X^2 \sim \chi^2_k (
chi-squared distribution In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squa ...
) * \lim_\tfrac \xrightarrow\ N(0,1) \, (
Normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
) *If X \sim N(0,1)\, then , X , \sim \chi_1 \, *If X \sim \chi_1\, then \sigma X \sim HN(\sigma)\, (
half-normal distribution In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution. Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the hal ...
) for any \sigma > 0 \, * \chi_2 \sim \mathrm(1)\, (
Rayleigh distribution In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom. The distribut ...
) * \chi_3 \sim \mathrm(1)\, (
Maxwell distribution Maxwell may refer to: People * Maxwell (surname), including a list of people and fictional characters with the name ** James Clerk Maxwell, mathematician and physicist * Justice Maxwell (disambiguation) * Maxwell baronets, in the Baronetage of ...
) * \, \boldsymbol_\, _2 \sim \chi_k , the
Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
of a standard normal random vector of with k dimensions, is distributed according to a chi distribution with k degrees of freedom *chi distribution is a special case of the
generalized gamma distribution The generalized gamma distribution is a continuous probability distribution with two shape parameters (and a scale parameter). It is a generalization of the gamma distribution which has one shape parameter (and a scale parameter). Since many dis ...
or the
Nakagami distribution The Nakagami distribution or the Nakagami-''m'' distribution is a probability distribution related to the gamma distribution. The family of Nakagami distributions has two parameters: a shape parameter m\geq 1/2 and a second parameter controlling ...
or the noncentral chi distribution *The mean of the chi distribution (scaled by the square root of n-1) yields the correction factor in the unbiased estimation of the standard deviation of the normal distribution.


See also

*
Nakagami distribution The Nakagami distribution or the Nakagami-''m'' distribution is a probability distribution related to the gamma distribution. The family of Nakagami distributions has two parameters: a shape parameter m\geq 1/2 and a second parameter controlling ...


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

* http://mathworld.wolfram.com/ChiDistribution.html {{DEFAULTSORT:Chi Distribution Continuous distributions Normal distribution Exponential family distributions