Background
Let be k independent, normally distributed random variables with means and variances . Then the random variable
is distributed according to the noncentral chi-squared distribution. It 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:
is sometimes called the noncentrality parameter. Note that some references define in other ways, such as half of the above sum, or its square root.
This distribution arises in multivariate statistics as a derivative of the multivariate normal distribution. While the central chi-squared distribution is the squared norm of a random vector with distribution (i.e., the squared distance from the origin of a point taken at random from that distribution), the non-central is the squared norm of a random vector with distribution. Here is a zero vector of length k, and is the identity matrix of size k.
Read more about this topic: Noncentral Chi-squared Distribution
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