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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?

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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.

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As far as I know there is no closed form solution. Bisection is a conservative way of solving it numerically, and coding up Newton's method should be no trouble, as all the constituent parts have explicit formulas.

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