Folded normal distribution

The folded normal distribution is a probability distribution derived from the normal distribution by taking the absolute value of a normally distributed random variable. It is also known as the half-normal distribution when the mean of the underlying normal distribution is zero. This distribution is useful in situations where only the magnitude of a variable is of interest, regardless of its direction.

The folded normal distribution has two parameters usually denoted by the Greek letters μ and σ. μ is the mean of the underlying normal distribution, and σ is the standard deviation of the underlying normal distribution. The probability density function (PDF) is given by:

The cumulative distribution function (CDF) is given by:

where Φ(·) is the CDF of the standard normal distribution.

As a reflection of its nature, the folded normal distribution is defined only for non-negative values (x ≥ 0). The scale parameter σ must be strictly positive to ensure a valid probability distribution.

The folded normal distribution finds applications in scenarios where we observe only the magnitude of a normally distributed quantity, particularly in physical measurements and error analysis.

• Signal processing applications use the folded normal distribution to model the envelope of narrow-band Gaussian noise.

• Reliability engineering employs this distribution to analyze deviations from target specifications when only the magnitude of deviation matters.

• Environmental science uses the folded normal to model pollutant concentrations where negative values are physically impossible.

• Quality control processes utilize this distribution when measuring absolute deviations from a target value.

• Metrology and calibration procedures employ the folded normal to analyze measurement uncertainties when only the magnitude of error is relevant.

• Wind speed modeling uses this distribution when the underlying wind components are normally distributed but only speed magnitude is measured.

When $\mu$ is zero (half-normal distribution), the skewness and kurtosis excess have constant values of approximately 0.995272 and 0.869177, respectively.

The folded normal distribution has several notable properties:

• When $\mu =0$, it reduces to the half-normal distribution

• The mode depends on both $\mu$ and $\sigma$ and occurs at $max(0,|\mu |-{\sigma }^2/|\mu |)$

• If $X$ follows a normal distribution with parameters $\mu$ and $\sigma$, then $|X|$ follows a folded normal distribution

• The half-normal distribution is a special case when $\mu =0$

The folded normal distribution is implemented by the FoldedNormalDistribution class. It has two constructors. The first constructor takes two arguments. The first argument is the mean of the underlying normal distribution, which acts as the location parameter. The second argument is the standard deviation of the underlying normal distribution and acts as the scale parameter. The following constructs the folded normal distribution with mean 6.8 and standard deviation 4.1:

var foldedNormal = new FoldedNormalDistribution(6.8, 4.1);

Dim foldedNormal = New FoldedNormalDistribution(6.8, 4.1)

No code example is currently available or this language may not be supported.

let foldedNormal = FoldedNormalDistribution(6.8, 4.1)

For details of the properties and methods common to all continuous distribution classes, see the topic on continuous distributions.

For more information on the folded normal distribution, refer to the following sources:

• Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 1", Chapter 13. Wiley Series in Probability and Statistics.

• Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). "Statistical Distributions", Chapter 33. Wiley Series in Probability and Statistics.

• John, S. (1982). "The Folded Normal Distribution: Two Methods of Estimation". Journal of Statistical Computation and Simulation.

• Leone, F. C., Nelson, L. S., & Nottingham, R. B. (1961). "The Folded Normal Distribution". Technometrics.

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