Truncated normal distribution

In probability and statistics, the truncated normal distribution is the probability distribution of a normally distributed random variable whose value is either bounded below or above (or both). The truncated normal distribution has wide applications in statistics and econometrics. For example, it is used to model the choice probability of a binary outcome in the probit model and to model censored data in the tobit model.

Definition

Suppose $X\; sim\; N(mu,\; sigma^\{2\})\; !$ has a normal distribution and lies within the interval . Then $X$ conditional on

:$f(x;mu,sigma,\; a,b)\; =\; frac\{frac\{1\}\{sigma\}phi(frac\{x\; -\; mu\}\{sigma\})\}\{Phi(frac\{b\; -\; mu\}\{sigma\})\; -\; Phi(frac\{a\; -\; mu\}\{sigma\})\; \},$

where $scriptstyle\{phi(cdot)\}$ is the probability density function of the standard normal distribution, $scriptstyle\{Phi(cdot)\}$ its cumulative distribution function, with the understanding that if $scriptstyle\{b=infty\}$, then $scriptstyle\{Phi(frac\{b\; -\; mu\}\{sigma\})\; =1\}$. ( And if $scriptstyle\{a=-infty\}$, then $scriptstyle\{Phi(frac\{a\; -\; mu\}\{sigma\})\; =0\}$).

Moments

Two sided truncation: :

One sided truncation:

:

where $alpha=(a-mu)/sigma,;\; lambda=phi(alpha)/\; [1-Phi(alpha)]\; ;\; ext\{and\}\; ;\; delta(alpha)\; =\; lambda(alpha)\; [lambda(alpha)-alpha]\; .$

See also

* Normal distribution
* Truncated distribution

References

*
* Norman L. Johnson and Samuel Kotz(1970). "Continuous univariate distributions-1", chapter 13. John Wiley & Sons.