# Statistics¶

## Distribution relationships¶

### Gamma & Wishart¶

Gamma(k,\theta) = Wishart(\nu, \sigma^2)\,,

where $k=\nu/2$ and $\theta=2\sigma^2$.

### Gamma & Chi-squared¶

• $\chi^2_k = Gamma(k/2, 2)$
• $c\chi^2_k = Gamma(k/2, 2c)$
• $\sum_{i=1}^n\chi^2_k(\lambda) = \chi^2_{nk}(n\lambda)$.

## Calculus¶

1. $\int_0^\infty e^{-x^2}dx$, see Impossible Integral.
2. Integral Inequality Absolute Value: $|\int^b_af(x)g(x) dx∣\le\int_a^b|f(x)|⋅|g(x)| dx$
3. $\sum \dfrac{1}{n\log n}$ and $\sum \dfrac{1}{n\log n\log(\log n)}$ diverges, while $\sum\dfrac{1}{x\log^2x}$ converges, refer to Infinite series … for more details.

## Simplex Volume¶

A simplex in $n$-dimensional Euclidean space is a convex solid with $n+1$ vertices, its volume is

V_n = \frac{1}{n!}h_nh_{n-1}\cdots h_1\,,

where $h_i$ is the distance between the $i$-th and the $i+1$-th vertices.

See Simplex Volumes and the Cayley-Menger Determinant for more details.

## Interesting Graphs¶

Cardioid (心形)：$r=2a(1+\cos\theta)$

## Confidence set for parameter vector in linear regression¶

Confidence set for parameter vector in linear regression

## Spherical and Elliptical Distributions¶

Spherical and Elliptical Distributions

## Subscript notation in expectations.¶

Always confused the risk function

\E_P[L(\theta(P),\hat\theta)]\,,

more clearly one for me is

R_T(P)=\E[L(P,T(X))]=\int_{\cal X}L(P,T(x))dP_X(x)

And one related question asked in Subscript notation in expectations