Question

Benford's Law states that the probability that the first decimal-digit of a raw data sample (from 1 to 9) is given

Answer 1

Answer:  P(4) = 0.097

                $\bold{\sum_{m=1}^9=1}$

Step-by-step explanation:

$P(m)=\log(m+1)-\log(m)\\\\.\qquad =\log\bigg(\dfrac{m+1}{m}\bigg)\qquad \text{(using rules for condensing logs)}\\\\\\P(4)=\log\bigg(\dfrac{4+1}{4}\bigg)\\\\\\.\qquad =\log\bigg(\dfrac{5}{4}\bigg)\\\\\\.\qquad =\large\boxed{0.097}$

$\sum_{m=1}^9\log\bigg(\dfrac{m+1}{m}\bigg)\\\\\\.\qquad =\log\bigg(\dfrac{2}{1}\bigg)+\log\bigg(\dfrac{3}{2}\bigg)+...\log\bigg(\dfrac{9}{8}\bigg)+\log\bigg(\dfrac{10}{9}\bigg)\\\\\\.\qquad =\log\bigg(\dfrac{2}{1}\bigg)\bigg(\dfrac{3}{2}\bigg)...\bigg(\dfrac{9}{8}\bigg)\bigg(\dfrac{10}{9}\bigg)\qquad \text{(using rules for condensing logs)}\\\\\\.\qquad =\log\bigg(\dfrac{10!}{9!}\bigg)\\\\\\.\qquad =\log(10)\\\\\\.\qquad =\large\boxed{1}$