I have a random variable $x$ that follows a lognormal disbution: $\log(x)\sim N(\mu, \sigma^2)$.
I want to truncate this distribution over some interval $[a, b]$ with $0<a<b<\infty$ while keeping $\mu$ and $\sigma$ unchanged.
Can someone explain how we do that or give a good reference that explains how to do it?
I am going to illustrate with the normal distribution because it involves less typing but the same can be done for the log normal although all the expressions will be messier.
Parameterise the truncated distribution with $\mu'$ and $\sigma'$. So
$f(x|\mu',\sigma')=B(a,b,\mu',\sigma')exp(-(x-\mu')^2/2\sigma'^2)$
Where $B$ is the new normalization $B=\frac{2}{\sqrt{2\pi\sigma'^2}(erf(\frac{b-\mu'}{\sqrt{2}\sigma'})-erf(\frac{a-\mu'}{\sqrt{2}\sigma'}))}$ (do check this.)
The new mean (call it M) is then given by a horrible looking expression involving $a,b,\mu',\sigma'$. Ditto for the variance (call it V).
Now set $M=\mu$ and $V=\sigma^2$ and solve for $\mu'$ and $\sigma'$ in terms of $a,b,\mu,\sigma$.
I am quite sure there will be no closed form expression for $\mu'$ and $\sigma'$.