Appendix

Put Appendix intro here.

Truncated normal distribution

The truncated normal distribution with parameters \(\mu\) and \(\sigma\) and lower-bound cutoff \(c_{lb}\) and upper-bound cutoff \(c_{ub}\) is simply the normal distribution of values of the random variable \(x\) defined only on the interval \(x\in[c_{lb}, c_{ub}]\) rather than on the full real line. And the probability distribution function values are upweighted by the probability (less than one) under the normal distribution on the interval \([c_{lb}, c_{ub}]\).

(1)\[\begin{split} \text{truncated normal:}\quad &f(x|\mu,\sigma,c_{lb},c_{ub}) = \frac{\phi(x|\mu,\sigma)}{\Phi(c_{ub}|\mu,\sigma) - \Phi(c_{ub}|\mu,\sigma)} \\ &\text{where}\quad \phi(x|\mu,\sigma) \equiv \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{x - \mu}{2\sigma^2}} \\ &\text{and}\quad \Phi(x|\mu,\sigma) \equiv \int_{-\infty}^x\phi(x|\mu,\sigma) dx\end{split}\]

The function \(\phi(x|\mu,\sigma)\) is the probability distribution function of the normal distribution with mean \(\mu\) and variance \(\sigma^2\). And the function \(\Phi(x|\mu,\sigma)\) is the cummulative distribution function of the normal distribution with mean \(\mu\) and variance \(\sigma^2\).

Footnotes

The footnotes from this appendix.