User:HighlyOrthodox/sandbox

Orthodox View

 * $$ \hat{p}, \hat{k}^{A}, \hat{k}^{B}, D^{A} $$

t distribution
$$ Z \sim N(0,1), \quad W \sim \chi^{2} (p) \Rightarrow  \frac{Z}{ \sqrt{W/p} } \sim t(p) $$

Easy to prove that the sample mean of Gaussian dist is an unbiased estimator, and s^2 is an unbiased estimator of the variance:


 * $$ E[ \bar{X} ] = E[ \frac{1}{n} \sum_{i} X_{i}  ] = \frac{1}{n } \sum_{i} \mu = \mu \;, $$


 * $$ E[ s^{2} ] = \sigma^{2} $$
 * $$ s^{2} = \frac{1}{n-1} \sum_{j} ( X_{j} - \bar{X} )^{2 } $$

numpy examples
pattern matching

If a matrix is normal, it can be normalized by a unitary matrix.


 * $$ \frac{1}{2 \pi i} \oint_{C} \frac{1 }{\zeta -z } d \zeta = 1 $$
 * $$ \int_{-1}^{1} \frac{g(x) }{\sqrt{1 - x^{2} } } dx \approx \sum_{i=1}^{N} w_{i} g(x_{i} )$$

Logistic growth model
$$ g(t) = \frac{ g_{max} }{ 1 + ( g_{max} /g_{0}  - 1  ) e^{-kt}  } $$