The Truncated Normal Distribution

The Truncated Normal DistributionJohn BurkardtDepartment of Scientific ComputingFlorida State jburkardt/ October 2014 AbstractThe Normal Distribution is a common model of randomness. Unlike the uniform Distribution , itproposes a most probable value which is also the mean, while other values occur with a probabilitythat decreases in a regular way with distance from the mean. This behavior is mathematically verysatisfying, and has an easily observed correspondence with many physical processes. One drawback ofthe Normal Distribution , however, is that it supplies a positive probability density to every value in therange ( ,+ ), although the actual probability of an extreme event will be very low. In many cases, itis desired to use the Normal Distribution to describe the random variation of a quantity that, for physicalreasons, must be strictly positive. A mathematically defensible way to preserve the main features of thenormal Distribution while avoiding extreme values involves thetruncated Normal Distribution , in whichthe range of definition is made finite at one or both ends of the interval.

between a randomly selected item and the mean. Assuming the mean is known, the variance is de ned as: var(ˆ()) = Z b a (x )2 ˆ(x)dx For the standard normal distribution, we have that var(˚(0;1;)) = 1. Note that the standard deviation of any distribution, represented by std(ˆ()), is simply the square root

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