(New page: Classic central limit Thm (Second Fundamental probabilistic): "The distribution of the average of a large number of samples from a distribution tends to be normal" let X1,X2,...,Xn be n ...) |
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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> | 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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| + | More generalization of central limit Thm. | ||
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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> | ||
Revision as of 11:01, 9 May 2010
Classic central limit Thm (Second Fundamental probabilistic):
"The distribution of the average of a large number of samples from a distribution tends to be normal"
let X1,X2,...,Xn be n independent and identically distributed variables (i.i.d) with finite mean $ \mu $ and finite variance $ \sigma^2>0 $.Then as n increases the distribution of $ \Sigma_{i=1}^n \frac{X_i} {n} $ approaches $ N(\mu,\frac {\sigma^2}{n}) $.
More precisely the random variable $ Z_n = \frac{\Sigma_{i=1}^n X_i - n \mu}{\sigma \sqrt{n}} $ has $ P(Z_n)\longrightarrow N(0,1) $ when $ n \longrightarrow \infty $
More generalization of central limit Thm.
let X1,X2,...,Xn be n independent variables
Xi has mean $ \mu_i $ & finite variance $ \sigma^2 > 0 $ ,i=1,2,...,n
Then $ Z_n = \frac{\Sigma_{i=1}^n X_i - \Sigma_{i=1}^n \mu_i} {\sqrt{\Sigma_{i=1}^n \sigma^2}} $ has $ P(Z_n)\longrightarrow N(\mu ,\Sigma) $ when $ n \longrightarrow \infty $
