I have an arbitrary number of independnet bivariate normal distributions with $\mu_i = [x_i,z_i]$ & $ \Sigma_i= \left(\begin{array}{cc} \sigma^2_{x_i} & \sigma^2_{x_i,z_i}\\\ \sigma^2_{x_i, z_i} & \sigma^2_{z_i} \end{array}\right) $

Where i is arbitrarily large

I want to take a linear combination of these bivariate normal distributions with weights $c = [c_1,...,c_i]$ where $\sum c_i = 1$

Obviously, the linear combination of $\mu_{full} = [\sum c_ix_i,\sum c_iz_i]$

However, I am not sure about the linear combination of the variance-covariance matrix.

Does anyone know how I can calculate this pooled & weighted variance-covariance? Thanks so much!

I have an arbitrary number of independnet bivariate normal distributions with $\mu_i = [x_i,z_i]$ & $ \Sigma_i= \left(\begin{array}{cc} \sigma^2_{x_i} & \sigma^2_{x_i,z_i}\\\ \sigma^2_{x_i, z_i} & \sigma^2_{z_i} \end{array}\right) $

Where i is arbitrarily large

I want to take a linear combination of these bivariate normal distributions with weights $c = [c_1,...,c_i]$ where $\sum c_i = 1$ & $c_i >0$

Obviously, the linear combination of $\mu_{mixture} = [\sum c_ix_i,\sum c_iz_i]$

However, I am not sure about the linear combination of the variance-covariance matrix.

Does anyone know how I can calculate this pooled & weighted variance-covariance? Looking for the variance-covariance matrix for the mixture distribution.

Thanks so much!

I have an arbitrary number of independnet bivariate normal distributions with $\mu_i = [x_i,z_i]$ & $ \Sigma_i= \left(\begin{array}{cc} \sigma^2_{x_i} & \sigma^2_{x_i,z_i}\\\ \sigma^2_{x_i, z_i} & \sigma^2_{z_i} \end{array}\right) $

Where i is arbitrarily large

I want to take a linear combination of these bivariate normal distributions with weights $c = [c_1,...,c_i]$ where $\sum c_i = 1$

Obviously, the linear combination of $\mu_i = [\sum c_ix_i,\sum c_iz_i]$

However, I am not sure about the linear combination of the variance-covariance matrix.

Does anyone know how I can calculate this pooled & weighted variance-covariance? Thanks so much!

I have an arbitrary number of independnet bivariate normal distributions with $\mu_i = [x_i,z_i]$ & $ \Sigma_i= \left(\begin{array}{cc} \sigma^2_{x_i} & \sigma^2_{x_i,z_i}\\\ \sigma^2_{x_i, z_i} & \sigma^2_{z_i} \end{array}\right) $

Where i is arbitrarily large

I want to take a linear combination of these bivariate normal distributions with weights $c = [c_1,...,c_i]$ where $\sum c_i = 1$

Obviously, the linear combination of $\mu_{full} = [\sum c_ix_i,\sum c_iz_i]$

However, I am not sure about the linear combination of the variance-covariance matrix.

Does anyone know how I can calculate this pooled & weighted variance-covariance? Thanks so much!