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| − | = Discriminant Functions For The Normal Density = | + | = Discriminant Functions For The Normal Density = |
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| − | Lets begin with the continuous univariate normal or Gaussian density. | + | Lets begin with the continuous univariate normal or Gaussian density. |
<div style="margin-left: 25em;"> | <div style="margin-left: 25em;"> | ||
| − | <math>f_x = \frac{1}{\sqrt{2 \pi} \sigma} \exp \left [- \frac{1}{2} \left ( \frac{x - \mu}{\sigma} \right)^2 \right ] </math> | + | <math>f_x = \frac{1}{\sqrt{2 \pi} \sigma} \exp \left [- \frac{1}{2} \left ( \frac{x - \mu}{\sigma} \right)^2 \right ] </math> |
| − | </div> | + | </div> |
| − | + | <br> for which ''the expected value'' of ''x'' is | |
| − | for which ''the expected value'' of ''x'' is | + | |
<div style="margin-left: 25em;"> | <div style="margin-left: 25em;"> | ||
| − | <math>\mu = \mathcal{E}[x] =\int\limits_{-\infty}^{\infty} xp(x)\, dx</math> | + | <math>\mu = \mathcal{E}[x] =\int\limits_{-\infty}^{\infty} xp(x)\, dx</math> |
| − | </div> | + | </div> |
and where the expected squared deviation or ''variance'' is | and where the expected squared deviation or ''variance'' is | ||
<div style="margin-left: 25em;"> | <div style="margin-left: 25em;"> | ||
| − | <math>\sigma^2 = \mathcal{E}[(x- \mu)^2] =\int\limits_{-\infty}^{\infty} (x- \mu)^2 p(x)\, dx</math> | + | <math>\sigma^2 = \mathcal{E}[(x- \mu)^2] =\int\limits_{-\infty}^{\infty} (x- \mu)^2 p(x)\, dx</math> |
| − | </div> | + | </div> |
| + | |||
| + | The univariate normal density is completely specified by two parameters; its mean ''μ '' and variance ''σ<sup>2</sup>''. Eq.1 f<sub>x</sub> can be written as ''N(μ,σ) | ||
| − | For the multivariate normal density in ''d'' dimensions, f<sub>x</sub> is written as | + | For the multivariate normal density in ''d'' dimensions, f<sub>x</sub> is written as |
<div style="margin-left: 25em;"> | <div style="margin-left: 25em;"> | ||
| − | <math>f_x = \frac{1}{(2 \pi)^ \frac{d}{2} |\boldsymbol{\Sigma}|^\frac{1}{2}} \exp \left [- \frac{1}{2} (\mathbf{x} -\boldsymbol{\mu})^t\boldsymbol{\Sigma}^{-1} (\mathbf{x} -\boldsymbol{\mu}) \right] </math> | + | <math>f_x = \frac{1}{(2 \pi)^ \frac{d}{2} |\boldsymbol{\Sigma}|^\frac{1}{2}} \exp \left [- \frac{1}{2} (\mathbf{x} -\boldsymbol{\mu})^t\boldsymbol{\Sigma}^{-1} (\mathbf{x} -\boldsymbol{\mu}) \right] </math> |
</div> | </div> | ||
Revision as of 16:53, 4 April 2013
Discriminant Functions For The Normal Density
Lets begin with the continuous univariate normal or Gaussian density.
$ f_x = \frac{1}{\sqrt{2 \pi} \sigma} \exp \left [- \frac{1}{2} \left ( \frac{x - \mu}{\sigma} \right)^2 \right ] $
for which the expected value of x is
$ \mu = \mathcal{E}[x] =\int\limits_{-\infty}^{\infty} xp(x)\, dx $
and where the expected squared deviation or variance is
$ \sigma^2 = \mathcal{E}[(x- \mu)^2] =\int\limits_{-\infty}^{\infty} (x- \mu)^2 p(x)\, dx $
The univariate normal density is completely specified by two parameters; its mean μ and variance σ2. Eq.1 fx can be written as N(μ,σ)
For the multivariate normal density in d dimensions, fx is written as
$ f_x = \frac{1}{(2 \pi)^ \frac{d}{2} |\boldsymbol{\Sigma}|^\frac{1}{2}} \exp \left [- \frac{1}{2} (\mathbf{x} -\boldsymbol{\mu})^t\boldsymbol{\Sigma}^{-1} (\mathbf{x} -\boldsymbol{\mu}) \right] $
