Given $N$ means and variances $\{\mu_1,\mu_2,....\mu_N\}$ , $\{\sigma_1^2,\sigma_2^2,....\sigma_N^2 \}$ ,and the fact that combined they make a gaussian mixture, how do I compute for that mixture $M$, its mean $\mu_M$ and variance $\sigma_M^2$ using the component means and variances?
Assume equal weight for each component.
A mixture PDF is $$f_M(x)=\frac{1}{N}\sum_{i=1}^N f_i(x)$$ Integrating for finding the mean yields: $$\mu_M=\int_{-\infty}^\infty xf_M(x)dx={1\over N}\sum_{i=1}^N\int_{-\infty}^\infty xf_i(x)dx={1\over N}\sum_{i=1}^N\mu_i$$
For the variance, let's find the second moments first: $$E[M^2]=\int_{-\infty}^\infty x^2f_M(x)dx={1\over N}\sum_{i=1}^N\int_{-\infty}^\infty x^2f_i(x)dx={1\over N}\sum_{i=1}^N(\mu_i^2+\sigma_i^2)$$ And, your variance will be $$ \sigma_M^2=E[M^2]-\mu_M^2$$