The folded normal distribution is a probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

 related to the normal distribution. Given a normally distributed random variable X with mean

Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

 μ and variance

Variance

In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

 σ², the random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

 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

Absolute value

In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

.

The probability density function

Probability density function

In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

 (PDF) is given by


The cumulative distribution function

Cumulative distribution function

In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

 (CDF) is given by

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

Alternatively, using the change of variables in the first integral and in the second integral, one can show that

where erf(x) is the error function

Error function

In mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics and partial differential equations...

, which is a standard function in many mathematical software packages. This expression reduces to the CDF of the half-normal distribution when μ = 0.

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 as the location and scale parameters of the new distribution.

Related distributions

• When μ = 0, the distribution of Y is a half-normal distribution
  Half-normal distribution
  The half-normal distribution is the probability distribution of the absolute value of a random variable that is normally distributed with expected value 0 and variance σ2. I.e...
  
  .
• (Y/σ) has a noncentral chi distribution with 1 degree of freedom and noncentrality equal to μ/σ.

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