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Concept
Loi normale repliée
Résumé
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 x = 0 is folded over by taking the absolute value. In the physics of heat conduction, the folded normal distribution is a fundamental solution of the heat equation on the half space; it corresponds to having a perfect insulator on a hyperplane through the origin.
The probability density function (PDF) is given by
for x ≥ 0, and 0 everywhere else. An alternative formulation is given by
where cosh is the cosine Hyperbolic function. It follows that the cumulative distribution function (CDF) is given by:
for x ≥ 0, where erf() is the error function. This expression reduces to the CDF of the half-normal distribution when μ = 0.
The mean of the folded distribution is then
or
where is the normal cumulative distribution function:
The variance then is expressed easily in terms of the mean:
Both the mean (μ) and variance (σ2) of X in the original normal distribution can be interpreted as the location and scale parameters of Y in the folded distribution.
The mode of the distribution is the value of for which the density is maximised. In order to find this value, we take the first derivative of the density with respect to and set it equal to zero. Unfortunately, there is no closed form. We can, however, write the derivative in a better way and end up with a non-linear equation
Tsagris et al. (2014) saw from numerical investigation that when , the maximum is met when , and when becomes greater than , the maximum approaches . This is of course something to be expected, since, in this case, the folded normal converges to the normal distribution. In order to avoid any trouble with negative variances, the exponentiation of the parameter is suggested.
Source officielle
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