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Lists and Tables

Lists
Variable Lists
Tables

A numbered list:

  1. One
  2. Two
  3. Three
    1. Three.a
    2. Three.b
    3. Three.c
  4. Four
    1. Four.a
      1. Four.a.i
      2. Four.a.ii
  5. Five

An unordered list:

  • First
  • Second
  • Third

A mixture of the two:

  1. 1
    • 1.a
      1. 1.a.1
      2. 1.a.2
    • 1.b
  2. 2
    • 2.a
    • 2.b
      1. 2.b.1
      2. 2.b.2
        • 2.b.2.a
        • 2.b.2.b

A Variable List

term 1

The definition of term 1

term 2

The definition of term 2

term 3

The definition of term 3

Here's a big table with code and other tricky things:

Table 1. Notes on the Implementation of the Beta Distribution

Function

Implementation Notes

pdf

f(x;α,β) = xα - 1 (1 - x)β -1 / B(α, β)

Implemented using ibeta_derivative(a, b, x).

cdf

Using the incomplete beta function ibeta(a, b, x)

cdf complement

ibetac(a, b, x)

quantile

Using the inverse incomplete beta function ibeta_inv(a, b, p)

quantile from the complement

ibetac_inv(a, b, q)

mean

a/(a+b)

variance

a * b / (a+b)^2 * (a + b + 1)

mode

(a-1) / (a + b + 2)

skewness

2 (b-a) sqrt(a+b+1)/(a+b+2) * sqrt(a * b)

kurtosis excess

beta_dist_kurtosis

kurtosis

kurtosis + 3

parameter estimation

alpha

from mean and variance

mean * (( (mean * (1 - mean)) / variance)- 1)

beta

from mean and variance

(1 - mean) * (((mean * (1 - mean)) /variance)-1)

The member functions estimate_alpha and estimate_beta

from cdf and probability x

and either alpha or beta

Implemented in terms of the inverse incomplete beta functions

ibeta_inva, and ibeta_invb respectively.

estimate_alpha

ibeta_inva(beta, x, probability)

estimate_beta

ibeta_invb(alpha, x, probability)



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