N Dim Normal Distribution

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