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In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both) It is essentially a mixture of a discrete distribution (constant 0) and a continuous distribution (a truncated gaussian distribution with interval ) as a result of censoring. The truncated normal distribution has wide applications in statistics and econometrics.

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The truncated normal distribution is an important example In probability theory, the rectified gaussian distribution is a modification of the gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier) [7] the tobit model employs truncated distributions

Other examples include truncated binomial at x=0 and truncated poisson at x=0.

A truncated mean or trimmed mean is a statistical measure of central tendency, much like the mean and median It involves the calculation of the mean after discarding given parts of a probability distribution or sample at the high and low end, and typically discarding an equal amount of both. In practice, if the fraction truncated is very small the effect of truncation might be ignored when analysing data For example, it is common to use a normal distribution to model data whose values can only be positive but for which the typical range of values is well away from zero.

The bates distribution is the distribution of the mean of n independent random variables, each of which having the uniform distribution on [0,1] The simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution This is a special case when and , and it is described by this probability density function (or density) [11] the variable ⁠ ⁠ has a mean of 0 and a variance and standard deviation of 1

The density has its peak at and inflection points at and ⁠ ⁠

Let follow an ordinary normal distribution,