User:BeyondNormality/Adhikari-Sarkar distribution (type 2)

Also known as the distribution of most significant digit, the probability mass function of the Adhikari-Sarkar distribution (type 2) is given by



\mathrm{AdhikariSarkar2}(x; n) \equiv \Pr(X = x) = \frac{(x+1)^{\left| n\right| }-x^{\left| n\right| }}{10^{\left| n\right| }-1} $$


 * $$ x = 1, 2, \dots ,9 $$
 * $$ n \in \mathbb{Z} - \{0\} $$

Cumulative Distribution Function


 * $$ F_X(x) = \frac{(x+1)^{\left| n\right| }-1}{10^{\left| n\right| }-1} $$

Expected Value


 * $$ \mathbb E(X) = -\frac{2^{\left| n\right| }+3^{\left| n\right| }+4^{\left| n\right| }+5^{\left| n\right| }+6^{\left| n\right| }+7^{\left| n\right| }+8^{\left| n\right| }+9^{\left| n\right| }-9\ 10^{\left| n\right| }+1}{10^{\left| n\right| }-1} $$