The PDF $f(x)$ of a non-negative random variable $x$ has the structure $$f(x)=\exp (a-bx-cx^{2})$$ where $a$, $b$ and $c$ are any model parameters. It is assumed that $c\ge 0$ so that $f(x)$ does not diverge for large $x$. Also given that the mean value $$\mu=\frac{e^{a}-b}{2c}$$ and $$\sigma^{2}=\frac{1-b\mu}{2c}-\mu^2$$
Develop a numerical scheme for finding the parameters $a$, $b$ and $c$ so that the PDF agrees with any given mean $\mu$ and $\sigma^2$.