N Dim Normal Distribution

N Dim Normal Distribution.md

• Normal Distribution.md
• If data points are vectors $x=(x_1,\dots ,x_n{)}^{\prime }$ and RVs X_i fulfill the Central Limit Theorem.md,
• PDF.md $p(x)=\frac{1}{(2\pi {)}^{n/2}det(\Sigma {)}^{\frac{1}{2}}}exp\left(-\frac{1}{2}(x-\mu {)}^{\prime }{\Sigma }^{-1}(x-\mu )\right)$
• $\mu$ is expectation $E[Covariance.md|(X_{1}, …, X_{n}|[X_{1}, …, X_{n}|X_{1}, …, X_{n})’]]''](X_{1}, …, X_{n})’](Covariance.md|(X_{1}, …, X_{n}|X_{1}, …, X_{n})''.md) matrix
• $\Sigma (i,j)=E[(X_i-E[X_i])(X_j-E[X_j])]$
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• $\hat{\mu }=\frac{1}{N}{\Sigma }_i{x}_i$ and $\hat{\Sigma }=\frac{1}{N-1}{\Sigma }_i(x_i-\hat{\mu })(x_i-\hat{\mu }{)}^{\prime }$