Transcription of The Gaussian distribution
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Example: barber
The Gaussian distribution
4 3 2 = 0, = 1 = 1, =1/2 = 0, = 2 Figure 1: Examples of univariate GaussianpdfsN(x; , 2).The Gaussian distributionProbably the most-important distribution in all of statistics is theGaussian distribution ,also calledthenormal Gaussian distribution arises in many contexts and is widely used formodeling continuous random probability density function of the univariate (one-dimensional) Gaussian distribution isp(x| , 2) =N(x; , 2) =1 Zexp( (x )22 2).The normalization constantZisZ= 2 parameters and 2specify the mean and variance of the distribution , respectively: =E[x]; 2= var[x].Figure 1 plots the probability density function for several sets of parameters( , 2). The distributionis symmetric around the mean and most of the density ( ) is contained within 3 of may extend the univariate Gaussian distribution to a distribution overd-dimensional vectors,producing a multivariate analog. The probablity density function of the multivariate Gaussiandistribution isp(x| , ) =N(x; , ) =1 Zexp( 12(x )> 1(x )).
Examining these equations, we can see that the multivariate density coincides with the univariate density in the special case when 2is the scalar ˙. Again, the vector speci˙es the mean of the multivariate Gaussian distribution. The matrix speci˙es the covariance between each pair of variables in x: = cov(x;x) = E (x )(x )>:
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