How can I generate truncated normal draws from truncated standard normal draws? Some random variables are transformations of a standard normal. For example, a draw from a normal density with mean $b$ and variance $s^2$ is obtained as$$e = b + s\epsilon$$ where $\epsilon$ is a draw of a standard normal.
I have a model and I estimate various parameters of the model using the maximum likelihood principle in MATLAB. I obtain draws from the truncated standard normal using a random number genertor in MATLAB. What I want to do is to obtain truncated normal draws using the already generated truncated standard normal draws.

*

*If $\upsilon$ is a draw from truncated standard normal, can I calculate$$e = b + s\upsilon$$ to obtain a draw from truncated normal density?


*Can I define $b$ and $s$ as parameters to be estimated in my maximum likelihood procedure so that I do not need to calculate them analytically beforehand?

The difficulty with the change in location and scale in the transform
$$e = b + s\upsilon$$is that, if $\upsilon$ is a truncated standard random variable restricted to the interval $(\alpha,\beta)$ then $e$ is a truncated non-standard random variable restricted to the interval $(b+s\alpha,b+s\beta)$. In other words, the truncated Normal distribution is a four parameter family rather than two.
Hence, if the target is a truncated $\mathcal N(b,s^2)$ distribution restricted to the interval $(\alpha,\beta)$, the indentity
\begin{align}\mathbb P(e\le x)&=\mathbb P(e-b\le x-b)\\
&=\mathbb P(\underbrace{s^{-1}\{e-b\}}_\text{truncated $\mathcal N(0,1)$}\le s^{-1}\{x-b\})\\
&=\dfrac{\Phi(s^{-1}\{x-b\})-\Phi(s^{-1}\{\alpha-b\})}
{\Phi(s^{-1}\{\beta-b\})-\Phi(s^{-1}\{\alpha-b\})}\\
\end{align}
This shows that the truncated standard normal must be restricted to the interval
$$(s^{-1}\{\alpha-b\},s^{-1}\{\beta-b\})$$
In conclusion,

to simulate a truncated $\mathcal N(b,s^2)$ distribution restricted to
the interval $(\alpha,\beta)$,

*

*Generate a truncated standard normal $X$ restricted to the interval $(s^{-1}\{\alpha-b\},s^{-1}\{\beta-b\})$

*Apply the transform $Y=b+sX$

