統計についてメモ

• http://www-cdf.fnal.gov/physics/statistics/statistics_recommendations.html
• INTEGRATION OF THE NORMAL DISTRIBUTION CURVE

$X=\pmatrix{ X_1\\ X_2\\ .\\ .\\ .\\ X_n\\}$is a set of observables. $\mu_i$ is a mean of $X_i$. Covariance matrix is defined as $$\Sigma = \pmatrix{ <(X_1-\mu_1)(X_1-\mu_1)> & & <(X_1-\mu_1)(X_2-\mu_2)> & ... & <(X_1-\mu_1)(X_n-\mu_n)> & \\ <(X_2-\mu_2)(X_1-\mu_1)> & & <(X_2-\mu_2)(X_2-\mu_2)> & ... & <(X_2-\mu_2)(X_n-\mu_n)> & \\ .\\ .\\ .\\ <(X_n-\mu_n)(X_1-\mu_1)> & & <(X_n-\mu_n)(X_2-\mu_2)> & ... & <(X_n-\mu_n)(X_n-\mu_n)> & \\ }$$
Note: diagonal term is equal to $\sigma_i^2$.
$\Sigma_{ij}>0$ : positive correlation.
$\Sigma_{ij}=0$ : no correlation.
$\Sigma_{ij}<0$ : negative correlation.
Correlation matrix is defined as $$\rho = (\rho_{ij}), \rho_{ij}=\frac{cov(X_i,X_j)}{\sigma_i \sigma_j}$$

$$f(x_1,x_2,...,x_n)=\frac{1}{(2\pi)^{n/2}\sqrt{\det(\Sigma))}}\exp(-\frac{1}{2}(x_1-\mu_1,...,x_n-\mu_n)\Sigma^{-1}(x_1-\mu_1,...,x_n-\mu_n)^T)$$
For two variable case,
$$f(x,y)=\frac{1}{2\pi\sigma_x\sigma_y\sqrt{1-\rho}}\exp[-\frac{1}{2(1-\rho^2)}(\frac{(x-\mu_x)^2}{\sigma_x^2}-\frac{2\rho(x-\mu_x)(y-\mu_y)}{\sigma_x\sigma_y}+\frac{(y-\mu_y)^2}{\sigma_y^2})]$$

• http://approximity.com/papers/ptfopt/node19.html