Let $f_1(x), \dots, f_n(x)$ be Gaussian density functions with different parameters, and $w_1, \dots, w_n$ be real numbers that sum-up to unity. Now the function $g(x) = \sum_i w_i f_i(x)$ is also a density function and I call it mixture-Gaussian density.
It is easy to calculate central moments (e.g. mean) of this distribution when we know the central moments of the underlying normal distributions, using linearity of integrals:
$$\int x^k g(x) dx = \int x^k \sum_i w_i f_i(x) dx = \sum_i w_i \int x^k f_i(x) dx$$
(please correct me if I am wrong).
How can I however calculate the quantiles of the new distribution (e.g. median)? Ideally I would like to get the quantile function, given quantile functions of the underlying normal distributions. Is there closed form solution? If not, what would be an efficient numerical solution?
Best Answer
Since $F(x)$ for normal distribution is strictly increasing, you can map from quantile function to the density function to obtain $\mu_i$ and $\sigma^2_i\ \forall i$. You obtain the entire density function of $g(x)$ (which depends on only two moments) using $w_i$, $\mu_i$ and $\sigma^2_i\ \forall i$. You could then map density function to the quantile function.