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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.

1.1 Mathematical De nition The standard normal distribution is a probability density function (PDF) de ned over the interval (1 ;+1). The function is often symbolized as ˚(0;1;x). It may be represented by the following formula: ˚(0;1;x) = 1 p 2ˇ e x 2 2 Like any PDF associated with a continuous variable, ˚(0;1;x) may be interpreted to ...

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