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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.

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I don't understand what you are asking for. Taking a random number generator and getting a sequence of means and variances, and then considering the Normal distribution with those parameters seems to satisfy your conditions but hardly is what you are looking for. Can you clarify a little bit? – gt6989b Oct 28 '13 at 19:30
@gt6989b I'm trying to generate continuous functions, so that I can integrate them. They need to be symmetric and normalize-able. Or better yet the algorithm would make them normal to begin with. – Loourr Oct 28 '13 at 19:40
So why is my suggestion not good (i.e. generate random $\{\mu_i, \sigma_i\}$ and each pair implies a Normal distribution provided $\sigma_i > 0$)? – gt6989b Oct 28 '13 at 19:58
@gt6989b it's not bad, I'm just not sure how to turn that into a continuous function – Loourr Oct 28 '13 at 20:34
up vote 1 down vote accepted

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

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

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