User:PAR/Work2: Difference between revisions

Revision as of 16:30, 17 May 2005

Revision as of 16:37, 17 May 2005

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In [[probability theory]] and [[statistics]], the '''chi distribution''' is a continuous [[probability distribution]]. It has one parameter <math>k</math> which specifies the number of degrees of freedom. The distribution usually arises when a k-dimensional vector has its orthogonal components independently distributed according to a standard [[normal distribution|normal]] distribution. The absolute value 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. The distribution of the square of the argument will follow the [[chi-square distribution]] with the same number of degrees of freedom.

== Properties ==

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:<math>f_k(x) = \frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</math>

where <math>erfi(z)</math> is the complex [[error function]]. The [[momentcumulative generatingdistribution function]] is given by:

:<math>F_k(x)=P(k/2,x^2/2)\,</math>

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

:<math>M(t)=M\left(\frac{k}{2},\frac{1}{2},\frac{t^2}{2}\right)+</math>