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Noncentral Chi-Squared Distribution 📂Probability Distribution

Noncentral Chi-Squared Distribution

Definition

The continuous probability distribution that has the probability density function as follows with respect to degrees of freedom $r > 0$ and non-centrality $\lambda \ge 0$ is called the Noncentral Chi-squared Distribution. $$ f(x) = {{ 1 } \over { 2 }} e^{- \left( x + \lambda \right) / 2 } \left( {{ x } \over { \lambda }} \right)^{k/4 - 1/2} I_{r/2 - 1} \left( \sqrt{\lambda x} \right) \qquad, x \in (0,\infty) $$


Explanation

As the name suggests, the noncentral chi-squared distribution is a generalization of the chi-squared distribution, meaning it applies to independent random variables $$ X_{k} \sim N \left( \mu_{k} , 1^{2} \right) \qquad , k = 1 , \cdots , r $$ that follow a normal distribution when they are defined as $\lambda = \sum_{k=1}^{r} \mu_{k}^{2}$. It implies the following meaning in $Y$. $$ Y = \sum_{k=1}^{r} X_{k}^{2} \sim \chi^{2} \left( r , \lambda \right) $$ That is, $\lambda \ne 0$ does not mean the chi-squared distribution itself, but indicates that the centers of the normal distributions being squared and summed are not $0$.

See also

Chi-squared Distribution

Noncentral F-distribution