Noncentral Chi Distribution - Bivariate Non-central Chi Distribution

Bivariate Non-central Chi Distribution

Let, be a set of n independent and identically distributed bivariate normal random vectors with marginal distributions, correlation, and mean vector and covariance matrix

$E(X_j)= \mu=(\mu_1, \mu_2)^T, \qquad \Sigma = \begin{bmatrix} \sigma_{11} & \sigma_{12} \\ \sigma_{21} & \sigma_{22} \end{bmatrix} = \begin{bmatrix} \sigma_1^2 & \rho \sigma_1 \sigma_2 \\ \rho \sigma_1 \sigma_2 & \sigma_2^2 \end{bmatrix},$

with positive definite. Define

$U = \left^{1/2}, \qquad V = \left^{1/2}.$

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.

Read more about this topic: Noncentral Chi Distribution

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