User:HighlyOrthodox/sandbox

Syntax

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

>>> a0=np.random.randint(low=-3, high=7, size=(4,4) )
>>> a0
array([[ 0,  4,  0,  1],
       [ 4,  4, -2,  4],
       [ 2, -1, -3,  4],
       [ 0,  6, -1,  5]])
>>> a0.sum(axis=1)
array([ 5, 10,  2, 10])

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

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