User:BeyondNormality/Aldous distribution

Also known as the distribution of the number of extreme points in the convex hull of a random sample, the probability mass function of the Aldous distribution is given by

$$ \begin{align} \mathrm{Aldous}(x) \equiv \Pr(X = x) & = 2^{x-3} \left[ \frac{\log ^{-2+x}(2)}{(-2+x)!} - \sum _{i=x-1}^{\infty } \frac{\log ^i(2)}{i!} \right] \\ & = 2^{x-3} \left[ \frac{\log ^{-2+x}(2)}{(-2+x)!} - \frac{2 (\Gamma (x-1)-\Gamma (x-1,\log (2)))}{\Gamma (x-1)} \right] \end{align}$$
 * $$ x = 3, 4, 5, \dots $$

Recurrence relation
 * $$ 4 \log (2) (\log (4)-x)  \Pr (x) + 2  \left(x^2-x (1+\log (2))-2 \log ^2(2)\right) \Pr (x+1) + x (-x+1+\log (4)) \Pr (x+2) =0 $$

Expected Value
 * $$ \mathbb E(X) = 4 $$

Variance
 * $$ \operatorname{Var}(X) = 8 \log (4)-10 $$

Moment Generating Function
 * $$ M_X(t)=\frac{e^{2 t} \left(4^{e^t} \left(e^t-1\right)+1\right)}{2 e^t-1} $$

Characteristic Function
 * $$ \varphi_X(t)=\frac{e^{2 i t} \left(4^{e^{i t}} \left(e^{i t}-1\right)+1\right)}{2 e^{i t}-1} $$

Probability Generating Function
 * $$ G(t) = \frac{\left(4^t (t-1)+1\right) t^2}{2 t-1} $$