The Truncated Normal Distribution

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The Truncated Normal DistributionJohn BurkardtDepartment of Scientific ComputingFlorida State jburkardt/ October 2014AbstractThe 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.

Because the standard normal distribution is symmetric about the origin, it is immediately obvious that mean(˚(0;1;)) = 0. The variance of a distribution ˆ(x), symbolized by var(ˆ()) is a measure of the average squared distance between a randomly selected item and the mean. Assuming the mean is known, the variance is de ned as:

  Standards, Distribution, Normal, Truncated normal distribution, Truncated