User:BeyondNormality/Adhikari-Sarkar distribution (type 1)

Also known as the distribution of most significant digit, the probability mass function of the Adhikari-Sarkar distribution (type 1) is given by



\mathrm{AdhikariSarkar1}(x; n) \equiv \Pr(X = x) = \frac{(x+1)^{\frac{1}{\left| n\right| }}-x^{\frac{1}{\left| n\right| }}}{10^{\frac{1}{\left| n\right| }}-1} $$


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

Cumulative Distribution Function


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

Expected Value


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

}}+7^{\frac{1}{\left| n\right| }}+8^{\frac{1}{\left| n\right| }}+9^{\frac{1}{\left| n\right| }}-9\ 10^{\frac{1}{\left| n\right| }}+1}{10^{\frac{1}{\left| n\right| }}-1} $$