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| In [[probability theory]] and [[statistics]], a '''normal variance-mean mixture''' with mixing probability density <math>g</math> is the continuous probability distribution of a random variable <math>Y</math> of the form
| | Andrew Berryhill is what his spouse loves to contact him and he completely digs that name. Kentucky is where I've always been living. Distributing manufacturing has been his occupation for some time. It's not a typical factor but what she likes performing is to play domino but she doesn't have the time recently.<br><br>My blog post ... clairvoyance; [http://test.jeka-nn.ru/node/129 jeka-nn.ru], |
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| :<math>Y=\alpha + \beta V+\sigma \sqrt{V}X,</math>
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| where <math>\alpha</math> and <math>\beta</math> are real numbers and <math>\sigma > 0</math> and random variables <math>X</math> and <math>V</math> are [[independence (probability theory)|independent]], <math>X</math> is [[normal distribution|normally distributed]] with mean zero and variance one, and <math>V</math> is [[continuous probability distribution|continuously distributed]] on the positive half-axis with [[probability density function]] <math>g</math>. The [[conditional distribution]] of <math>Y</math> given <math>V</math> is thus a normal distribution with mean <math>\alpha + \beta V</math> and variance <math>\sigma^2 V</math>. A normal variance-mean mixture can be thought of as the distribution of a certain quantity in an inhomogeneous population consisting of many different normal distributed subpopulations. It is the distribution of the position of a [[Wiener process]] (Brownian motion) with drift <math>\beta</math> and infinitesimal variance <math>\sigma^2</math> observed at a random time point independent of the Wiener process and with probability density function <math>g</math>. An important example of normal variance-mean mixtures is the [[generalised hyperbolic distribution]] in which the mixing distribution is the [[generalized inverse Gaussian distribution]].
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| The probability density function of a normal variance-mean mixture with [[Mixture density|mixing probability density]] <math>g</math> is
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| :<math>f(x) = \int_0^\infty \frac{1}{\sqrt{2 \pi \sigma^2 v}} \exp \left( \frac{-(x - \alpha - \beta v)^2}{2 \sigma^2 v} \right) g(v) \, dv</math> | |
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| and its [[moment generating function]] is
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| :<math>M(s) = \exp(\alpha s) \, M_g \left(\beta s + \frac12 \sigma^2 s^2 \right),</math>
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| where <math>M_g</math> is the moment generating function of the probability distribution with density function <math>g</math>, i.e.
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| :<math>M_g(s) = E\left(\exp( s V)\right) = \int_0^\infty \exp(s v) g(v) \, dv.</math>
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| ==See also==
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| :*[[Normal-inverse Gaussian distribution]]
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| ==References==
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| O.E Barndorff-Nielsen, J. Kent and M. Sørensen (1982): "Normal variance-mean mixtures and z-distributions", ''International Statistical Review'', 50, 145–159.
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| [[Category:Continuous distributions]]
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| [[Category:Compound distributions]]
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Andrew Berryhill is what his spouse loves to contact him and he completely digs that name. Kentucky is where I've always been living. Distributing manufacturing has been his occupation for some time. It's not a typical factor but what she likes performing is to play domino but she doesn't have the time recently.
My blog post ... clairvoyance; jeka-nn.ru,