# 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 $X \in (a,b), \; -\infty \leq a < b \leq \infty$. Then X follows a truncation normal distribution with probability density function

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

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

If $X \sim N(\mu, \sigma^{2})\!$ has a normal distribution and b a constant, then

$E(X|X>b) = \mu +\sigma\lambda(\alpha), \; \; Var(X|X>b) = \sigma^2[1-\delta(\alpha)],\!$

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

## Reference

• Greene, William H. (2003). Econometric Analysis (5th ed.). Prentice Hall. ISBN 0-13-066189-9.