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Classic central limit Thm (Second Fundamental probabilistic):
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[[Category:2010 Spring ECE 662 mboutin]]
  
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"The distribution of the average of a large number of samples from a distribution tends to be normal"
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=Details of Lecture 8, [[ECE662]] Spring 2010=
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In Lecture 8, we justified the commonly made assumption that the features are normally distributed with the Central Limit Theorem. We then discussed the probability of error when using Bayes decision rule. More precisely, we obtained the Chernoff Bound and the Bhattacharrya bound for the probability of error.
  
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let X1,X2,...,Xn be n independent and identically distributed variables (i.i.d) with finite mean <math>\mu</math> and finite variance <math>\sigma^2>0</math>.Then as n increases the distribution of <math>\Sigma_{i=1}^n \frac{X_i} {n}</math> approaches <math>N(\mu,\frac {\sigma^2}{n})</math>.
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Note for this lecture can be found [[noteslecture8ECE662S10|here]].  
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More precisely the random variable <math>Z_n = \frac{\Sigma_{i=1}^n X_i - n \mu}{\sigma \sqrt{n}}</math> has <math>P(Z_n)\longrightarrow N(0,1)</math> when <math>n \longrightarrow \infty</math>
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Previous: [[Lecture7ECE662S10|Lecture 7]]
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Next: [[Lecture9ECE662S10|Lecture 9]]
  
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More generalization of central limit Thm.
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[[ 2010 Spring ECE 662 mboutin|Back to 2010 Spring ECE 662 mboutin]]
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let X1,X2,...,Xn be n independent variables
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Xi has mean <math>\mu_i</math> & finite variance <math>\sigma^2 > 0</math> ,i=1,2,...,n
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Then <math>Z_n = \frac{\Sigma_{i=1}^n X_i - \Sigma_{i=1}^n \mu_i} {\sqrt{\Sigma_{i=1}^n \sigma^2}}</math> has <math>P(Z_n)\longrightarrow N(\mu ,\Sigma)</math> when <math>n \longrightarrow \infty</math>
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Latest revision as of 09:09, 11 May 2010


Details of Lecture 8, ECE662 Spring 2010

In Lecture 8, we justified the commonly made assumption that the features are normally distributed with the Central Limit Theorem. We then discussed the probability of error when using Bayes decision rule. More precisely, we obtained the Chernoff Bound and the Bhattacharrya bound for the probability of error.

Note for this lecture can be found here.


Previous: Lecture 7 Next: Lecture 9

Back to 2010 Spring ECE 662 mboutin

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