Chapter 6.1, Problem 22E

(a)

To determine

To calculate: the standard deviation

Answer to Problem 22E

  $\sigma =2.58$

Explanation of Solution

Given:

First digit	1	2	3	4	5	6	7	8	9
Probability	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$9$}\right.$	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$9$}\right.$	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$9$}\right.$	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$9$}\right.$	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$9$}\right.$	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$9$}\right.$	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$9$}\right.$	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$9$}\right.$	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$9$}\right.$

Formula used:

  $\mu ={\displaystyle \sum yP(y)}$

  $\sigma =\sqrt{{\displaystyle \sum {\left(x-\mu \right)}^2\text{P}\left(x\right)}}$

Calculation:

  $\begin{array}{c}\mu ={\displaystyle \sum yP(y)}\\ =1\times \frac{1}{9}+1\times \frac{1}{9}+1\times \frac{1}{9}+2\times \frac{1}{9}+3\times \frac{1}{9}+4\times \frac{1}{9}+5\times \frac{1}{9}+6\times \frac{1}{9}+7\times \frac{1}{9}+8\times \frac{1}{9}+9\times \frac{1}{9}\\ =\frac{1+2+3+4+5+6+7+8+9}{9}\\ =\frac{45}{9}\\ =5\end{array}$

  $\begin{array}{c}\sigma =\sqrt{{\displaystyle \sum {\left(x-\mu \right)}^2\text{P}\left(x\right)}}\\ {\sigma }^2={\left(1-5\right)}^2\times \frac{1}{9}+{\left(2-5\right)}^2\times \frac{1}{9}+{\left(3-5\right)}^2\times \frac{1}{9}\\ +{\left(4-5\right)}^2\times \frac{1}{9}+{\left(5-5\right)}^2\times \frac{1}{9}+{\left(6-5\right)}^2\times \frac{1}{9}\\ +{\left(7-5\right)}^2\times \frac{1}{9}+{\left(8-5\right)}^2\times \frac{1}{9}+{\left(9-5\right)}^2\times \frac{1}{9}\\ =\frac{20}{3}\\ \sigma =\sqrt{{\sigma }^2}\\ =\sqrt{\frac{20}{3}}\\ =2.58\end{array}$

(b)

To determine

To Explain: that to detect fraud standard deviation is good or not

Answer to Problem 22E

Not good

Explanation of Solution

Not good, because by standard deviation by Ben ford’s law is 2.46 and by uniform distribution is 2.58. It is very close so it is hard to distinguish