Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Noncentral chi distribution
Summary
In probability theory and statistics, the noncentral chi distribution is a noncentral generalization of the chi distribution. It is also known as the generalized Rayleigh distribution.
If are k independent, normally distributed random variables with means and variances , then the statistic
is distributed according to the noncentral chi distribution. The noncentral chi distribution has two parameters: which specifies the number of degrees of freedom (i.e. the number of ), and which is related to the mean of the random variables by:
The probability density function (pdf) is
where is a modified Bessel function of the first kind.
The first few raw moments are:
where is a Laguerre function. Note that the 2th moment is the same as the th moment of the noncentral chi-squared distribution with being replaced by .
Let , be a set of n independent and identically distributed bivariate normal random vectors with marginal distributions , correlation , and mean vector and covariance matrix
with positive definite. Define
Then the joint distribution of U, V is central or noncentral bivariate chi distribution with n degrees of freedom.
If either or both or the distribution is a noncentral bivariate chi distribution.
If is a random variable with the non-central chi distribution, the random variable will have the noncentral chi-squared distribution. Other related distributions may be seen there.
If is chi distributed: then is also non-central chi distributed: . In other words, the chi distribution is a special case of the non-central chi distribution (i.e., with a non-centrality parameter of zero).
A noncentral chi distribution with 2 degrees of freedom is equivalent to a Rice distribution with .
If X follows a noncentral chi distribution with 1 degree of freedom and noncentrality parameter λ, then σX follows a folded normal distribution whose parameters are equal to σλ and σ2 for any value of σ.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.