# Approximate linear density function for a normal distribution

I'm working on implementing Order Preserving Encryption for Numeric Data, and part of the algorithm includes approximating density of the distribution as a linear density function $f(p) = qp+r$ where $p$ exists in the distribution.

From the article:

If a distribution over $[0,ph)$ has the density function $qp + r$, where $p \in [0,ph)$, then for any constant $z > 0$, the mapping function $M(p) = z(\frac{q}{2r}p^2 + p)$ will yield a uniformly distributed set of values.

The algorithm splits up the distribution into a set of "buckets", and a unique density function is needed for each bucket.

If I'm working with a normal distribution over $[0,100)$ with $\mu=65$ and $\sigma=10$, split into buckets $[0,40)$, $[40, 65)$, $[65, 80)$, and $[80,100)$, how can I produce a density function for each bucket?

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