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## Lists and Tables

### Lists

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

### Variable Lists

A Variable List

term 1

  The definition of term 1

term 2

  The definition of term 2

term 3

  The definition of term 3

### Tables

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 |
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)`