Moments
Two sided truncation:
![\operatorname{Var}(X \mid a<X<b) = \sigma^2\left[1+\frac{\frac{a-\mu}{\sigma}\phi(\frac{a-\mu}{\sigma})-\frac{b-\mu}{\sigma}\phi(\frac{b-\mu}{\sigma})}{\Phi(\frac{b-\mu}{\sigma})-\Phi(\frac{a-\mu}{\sigma})}
-\left(\frac{\phi(\frac{a-\mu}{\sigma})-\phi(\frac{b-\mu}{\sigma})}{\Phi(\frac{b-\mu}{\sigma})-\Phi(\frac{a-\mu}{\sigma})}\right)^2\right]\!](http://upload.wikimedia.org/math/d/2/1/d2178df330e9c5dab435c2ca696ba4e7.png)
One sided truncation (upper tail):
where and .
One sided truncation (lower tail):
where
Barr and Sherrill (1999) give a simpler expression for the variance of one sided truncations. Their formula is in terms of the chi-square CDF, which is implemented in standard software libraries. Bebu and Mathew (2009) provide formulas for (generalized) confidence intervals around the truncated moments.
Read more about this topic: Truncated Normal Distribution
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