User:BeyondNormality/Banach distribution

The probability mass function of the Banach distribution is



\mathrm{Banach}(x; n) \equiv \Pr(X = x) = 2^{x-2 n} \binom{2 n-x}{n} $$


 * $$ x = 0, 1, \dots, n $$
 * $$ n \in \mathbb{N}_0  $$

Recurrence Relation
 * $$ 2(n-x) \Pr (x)+(x-2 n) \Pr (x+1)=0 $$

Cumulative Distribution Function

F_X(x) = 4^{-n} \left(\binom{2 n}{n} \, _2F_1(1,-n;-2 n;2)-2^{x+1} \binom{2 n-x-1}{n} \, _2F_1(1,-n+x+1;-2 n+x+1;2)\right) $$

Expected Value
 * $$ \mathbb E(X) = 4^{-n} (2 n+1) \binom{2 n}{n}-1 $$

Moment Generating Function
 * $$ M_X(t) = 4^{-n} \binom{2 n}{n} \, _2F_1\left(1,-n;-2 n;2 e^t\right) $$

Characteristic Function
 * $$ \varphi_X(t) = 4^{-n} \binom{2 n}{n} \, _2F_1\left(1,-n;-2 n;2 e^{i t}\right) $$

Probability Generating Function
 * $$ G(t) = 4^{-n} \binom{2 n}{n} \, _2F_1(1,-n;-2 n;2 t) $$