Question

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

Pm = log (m +1) - log m. That is, about (100P)% of the data can be expected to have m as the first digit. Complete parts a and b below.
a. What percent of the data can be expected to have 4 as the first digit?
Pa =
(Round to three decimal places as needed.)
b. Find P1+P2 + ... + P9. Interpret your result.
Py + P2 + ... +P9 =
(Type an integer or a decimal.)​

Answer 1

Answer: P(4) = 0.097

`\bold{\sum_(m=1)^9=1}`

Explanation:

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

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