formulasearchengine - User contributions [en]

Landau quantization

2014-01-03T16:52:03Z

31.33.159.210: /* Derivation */

{{Probability distribution
| name =
| type = density
| pdf_image = [[Image:Folded normal pdf.svg|350px|Probability density function for the folded-normal distribution]] <math>\mu=1,\sigma=1</math>
| cdf_image = [[Image:Folded normal cdf.svg|350px|Cumulative distribution function for the normal distribution]] <math>\mu=1,\sigma=1</math>
| notation =
| parameters = {{nowrap|''μ'' ∈ '''R'''}} — ([[location parameter|location]]) {{nowrap|''σ''2 > 0}} — ([[scale parameter|scale]])
| support = {{nowrap|''x'' ∈ [0,<math>\infty</math>)}}
| pdf = (see article)
| cdf = (see article)
| mean = (see article)
| median =
| mode =
| variance = (see article)
| skewness =
| kurtosis = 
| entropy =
| mgf =
| char =
| fisher =
}}

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 [[probability density function]] (PDF) is given by

:<math>f(x;\mu,\sigma)=
\frac{1}{\sigma\sqrt{2\pi}} \, \exp \left( -\frac{(-x-\mu)^2}{2\sigma^2} \right)
+ \frac{1}{\sigma\sqrt{2\pi}} \, \exp \left( -\frac{(x-\mu)^2}{2\sigma^2} \right)\qquad(x \ge 0\,)</math>

The [[cumulative distribution function]] (CDF) is given by

: <math>F_Y(y; \mu, \sigma) = \int_0^y \frac{1}{\sigma\sqrt{2\pi}} \, \exp \left( -\frac{(-x-\mu)^2}{2\sigma^2} \right)\, dx
+ \int_0^{y} \frac{1}{\sigma\sqrt{2\pi}} \, \exp \left( -\frac{(x-\mu)^2}{2\sigma^2} \right)\, dx.</math>

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

: <math>F_Y(y; \mu, \sigma) = \int_{-\mu/\sigma}^{(y-\mu)/\sigma} \frac{1}{\sqrt{2\pi}} \, \exp \left(-\frac{1}{2}\left(z + \frac{2\mu}{\sigma}\right)^2\right) dz
+ \int_{-\mu/\sigma}^{(y-\mu)/\sigma} \frac{1}{\sqrt{2\pi}} \, \exp \left( -\frac{z^2}{2} \right) dz.
</math>

Alternatively, using the change of variables <math>z = -(x+\mu)/\sqrt{2}\sigma</math> in the first integral and <math>z = (x-\mu)/\sqrt{2}\sigma</math> in the second integral, one can show that

: <math>F_Y(y; \mu, \sigma) = \frac{1}{2}\left[ \mbox{erf}\left(\frac{y+\mu}{\sqrt{2}\sigma}\right) + \mbox{erf}\left(\frac{y-\mu}{\sqrt{2}\sigma}\right)\right],
</math>
where erf(x) is the [[error function]], 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

: <math>E(y) = \sigma \sqrt{2/\pi} \exp(-\mu^2/2\sigma^2) + \mu\left[1-2\Phi(-\mu/\sigma)\right],</math>

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

The variance is given by

: <math>\operatorname{Var}(y) = \mu^2 + \sigma^2 - \left\{ \sigma \sqrt{2/\pi} \exp(-\mu^2/2\sigma^2) + \mu\left[1-2\Phi(-\mu/\sigma)\right] \right\}^2. </math>

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]].
* <math>(Y/\sigma)^2</math> has a [[noncentral chi-squared distribution]] with 1 degree of freedom and noncentrality equal to <math> (\mu/\sigma )^2</math>.

== See also ==
* [[Folded cumulative distribution]]

== References==

* {{cite journal | author=Leone FC, Nottingham RB, Nelson LS | year = 1961
| title = The Folded Normal Distribution
| journal = Technometrics | volume = 3 | issue = 4 | pages = 543–550
| doi = 10.2307/1266560 | jstor=1266560 | publisher=Technometrics, Vol. 3, No. 4
}}
* {{cite journal | author=Johnson NL | year = 1962
| title = The folded normal distribution: accuracy of the estimation by maximum likelihood
| journal = Technometrics | volume = 4 | issue = 2 | pages = 249–256
| doi = 10.2307/1266622 | jstor=1266622 | publisher=Technometrics, Vol. 4, No. 2
}}
* {{cite journal | author=Nelson LS | year = 1980
| title = The Folded Normal Distribution
| journal = J Qual Technol | volume = 12 | issue = 4 | pages = 236–238
| doi =
}}
* {{cite journal | author=Elandt RC | year = 1961
| title = The folded normal distribution: two methods of estimating parameters from moments
| journal = Technometrics | volume = 3 | issue = 4 | pages = 551–562
| doi = 10.2307/1266561 | jstor=1266561 | publisher=Technometrics, Vol. 3, No. 4
}}
* {{cite journal | author=Lin PC | year = 2005
| title = Application of the generalized folded-normal distribution to the process capability measures
| journal = Int J Adv Manuf Technol | volume = 26 | pages = 825–830
| doi = 10.1007/s00170-003-2043-x | issue=7–8
}}

{{ProbDistributions|continuous-semi-infinite}}

[[Category:Probability distributions]]
[[Category:Continuous distributions]]
[[Category:Normal distribution]]

S-matrix

2012-07-26T12:50:41Z

31.33.159.210: /* Use of S-matrices */

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Bloch oscillations

2012-07-25T06:40:17Z

31.33.159.210:

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