algorithm for generating interesting normal symmetric continuous functions

Question

Does any one know of, or can think of an algorithm which generates arbitrarily many symmetric normally distributed continuous functions? And when I say interesting I mean more complex distributions then an offset bell curve.

Answer 1

Clarifying our discussion in the comments.

Consider generating a sequence of real pairs $\left(\mu_k, \sigma_k\right)_{k=1}^N$ with $\sigma_k > 0$ for each $k$.

Then, for each $k$, consider the pdf of the $\mathcal{N} \left(\mu_k, \sigma_k \right)$ distribution, which is given by

$$ f(x) = \frac{\exp \left( -\frac{1}{2} \left(\frac{x-\mu_k}{\sigma_k}\right)^2 \right)} {\sigma_k \sqrt{2\pi}}. $$