Definition of Noncentral Chi-Squared Distribution

Definition: If n random values z₁, z₂, ..., z_n are drawn from normal distributions with known nonzero means and constant variance, then squared, and summed, the resulting statistic is said to have a noncentral chi-squared distribution with n degrees of freedom:

z₁² + z₂² + ... + z_n²) ~ X²_{(n, q)}

This is a two-parameter family of distributions. Parameter n is conventionally labeled the degrees of freedom of the distribution. Parameter q is the noncentrality parameter. It is related to the means m_i and variance s² of the normal distributions thus: q=(sum for i=1 to n) of (m_i² / s²).

The mean of a distribution that is X²_{(n, q)} is (n+q). The variance of that distribution is (2n+4q). (Econterms)

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