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Statistics

Distribution relationships

Gamma & Wishart

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

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

Refer to ARPM: Wishart and Gamma distribution

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(\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)

Bootstrap samples

  1. How many different bootstrap samples are there?

Confidence set for parameter vector in linear regression

Confidence set for parameter vector in linear regression

Order statistics

  1. Expected value of the k-th order statistic from uniform random variables

Spherical and Elliptical Distributions

Spherical and Elliptical Distributions

Wishart Distribution

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