User:BeyondNormality/Adhikari-Sarkar distribution (type 3)

Also known as the distribution of most significant digit, the probability mass function of the Adhikari-Sarkar distribution (type 3) is given by



\mathrm{AdhikariSarkar3}(x; m) \equiv \Pr(X = x) = \text{coefficient of } t^{m-1} \text{ in } \frac{(x+1)^{1-t}-x^{1-t}}{\left(10^{1-t}-1\right) (1-t)} $$


 * $$ x = 1, 2, \dots ,9 $$
 * $$ m \in \mathbb{N} $$
 * $$ \left| t \right| < 1 $$

Probability Mass Function


 * $$ \Pr(X = x) = \frac{(-1)^{-1+m}}{(-1+m)!}\sum _{r=1}^{\infty } \int_{\frac{x}{10^r}}^{\frac{x+1}{10^r}} \log ^{-1+m}(y) \, dy $$