Folded normal distribution

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The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable X with mean μ and variance σ2, the random variable Y = |X| has a folded normal distribution. Such a case may be encountered if only the magnitude of some variable is recorded, but not its sign. The distribution is called Folded because probability mass to the left of the x = 0 is "folded" over by taking the absolute value.

The cumulative distribution function (CDF) is given by

Using the change-of-variables z = (x − μ)/σ, the CDF can be written as

The expectation is then given by

where Φ(•) denotes the cumulative distribution function of a standard normal distribution.

The variance is given by

Both the mean, μ, and the variance, σ2, of X can be seen to location and scale parameters of the new distribution.

Related distributions

References

  • Leone FC, Nottingham RB, Nelson LS (1961). "The Folded Normal Distribution". Technometrics. 3 (4): 543–550. doi:10.2307/1266560.
  • Johnson NL (1962). "The folded normal distribution: accuracy of the estimation by maximum likelihood". Technometrics. 4 (2): 249–256. doi:10.2307/1266622.
  • Nelson LS (1980). "The Folded Normal Distribution". J Qual Technol. 12 (4): 236–238.
  • Elandt RC (1961). "The folded normal distribution: two methods of estimating parameters from moments". Technometrics. 3 (4): 551–562. doi:10.2307/1266561.
  • Lin PC (2005). "Application of the generalized folded-normal distribution to the process capability measures". Int J Adv Manuf Technol. 26: 825–830. doi:10.1007/s00170-003-2043-x.

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