This function is used to convert Halton draws to the specified distribution.
The function can be used to generate random draws for use in random parameter
models, generating Halton-based pseudo-random draws for specified
distributions, etc.
The distributions generated all use the `mean` ($$\mu$$) and `sdev`
($$\sigma$$) parameters to generate the random draws. The density
functions for the distributions are as follows:
The Normal distribution is:
\(f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}
\exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right)\)
The Lognormal distribution is:
\(f(x) =
\frac{1}{x\sigma\sqrt{2\pi}}
\exp\left(-\frac{(\log(x) - \mu)^2}{2\sigma^2}\right)\)
The Triangular distribution is (note that this is a symmetrical triangular
distribution where $$\mu$$ is the median and $$\sigma$$ is the
half-width):
\(f(x) = \begin{cases}
\frac{(x - \mu + \sigma)}{\sigma^2}, & \text{for } \mu -
\sigma \leq x \leq \mu \\
\frac{(\mu + \sigma - x)}{\sigma^2}, & \text{for }
\mu < x \leq \mu + \sigma \\0, & \text{otherwise}\end{cases}\)
The Uniform distribution is (note that \(\mu\) is the midpoint and
\(\sigma\) is the half-width):
\(f(x) = \frac{1}{(\beta_{\mu}+\beta_{\sigma}) -
(\beta_{\mu}-\beta_{\sigma})}=\frac{1}{2\beta_{\sigma}}\)
The Gamma distribution is based on $$\mu = \frac{\alpha}{\beta}$$ and
$$\sigma^2 = \frac{\alpha}{\beta^2}$$:
\(f(x) =
\frac{\left(\frac{\mu}{\sigma^2}\right)^
{\frac{\mu^2}{\sigma^2}}}{\Gamma\left(\frac{\mu^2}{\sigma^2}\right)}
x^{\frac{\mu^2}{\sigma^2} - 1} e^{-\frac{\mu}{\sigma^2} x}\)