Chapter 4, Problem 4.11P

The random variable X is said to follow the distribution of Benford’s Law if $P\left(X=i\right)={\log }_{10}\left(\frac{i+1}{i}\right),i=1,2,3,\mathrm{....},9$

It has been shown to be a good fit for the distribution of the first digit of many real life data values.

a. Verify that the preceding is a probability mass function by showing that ${\displaystyle {\sum }_{i=1}^{9}P\left(X=i\right)=1}$.

b. Find $P\left(X\le j\right)$.