Given a distribution $F$ defined as a linear combination of Gaussian distributions:

$F = \sum_{i=1}^n C_i*N(\mu_i,\sigma_i)$ with $\sum_{i=1}^n C_i = 1$

I want to find a Gaussian function $Q = a*e^{\frac{(x-b)^2}{2c^2}}$ such that $Q(x) \geq F(x)$ $\; \forall x \in \mathbb{R}$, and $a$ is minimal.

Some context: I will be using $Q$ as envelope function/proposal distribution ($\frac{Q}{a}$) in the rejection sampling method, to generate samples from $F$. I do realize it is easy to directly draw samples from $F$, but I wish to analyze/compare different monte carlo methods on this simple example. I was wondering whether a way to determine an optimal (or good) Q existed in general for this $F$. In principle I only require a few example cases, so if for some particular linear combination of Gaussian distributions (e.g. $n = 2$ or $C_i = C_j$ $\; \forall i,j$), determining the optimal $Q$ is straightforward that would be of interest too.

Thank you for your help

Given a distribution $F$ defined as a linear combination of Gaussian distributions:

$F = \sum_{i=1}^n C_i*N(\mu_i,\sigma_i)$ with $\sum_{i=1}^n C_i = 1$

I want to find a Gaussian function $Q = a*e^{\frac{(x-b)^2}{2c^2}}$ such that $Q(x) \geq F(x)$ $\; \forall x \in \mathbb{R}$, and $a$ is minimal.

Some context: I will be using $Q$ as envelope function/proposal distribution ($\frac{Q}{a}$) in the rejection sampling method, to generate samples from $F$. I do realize it is easy to directly draw samples from $F$, but I wish to analyze/compare different monte carlo methods on this simple example. I was wondering whether a way to determine an optimal (or good) $Q$ existed in general for this $F$. In principle I only require a few example cases, so if for some particular linear combination of Gaussian distributions (e.g. $n = 2$ or $C_i = C_j$ $\; \forall i,j$), determining the optimal $Q$ is straightforward that would be of interest too.

Thank you for your help