Chapter 6.1, Problem 16E

To determine

To find: the standard deviation of Y and explain it

Answer to Problem 16E

  $\sigma =1.769$

Explanation of Solution

Given:

Days(Y)	0	1	2	3	4	5	6	7
Probability	0.68	0.05	0.07	0.08	0.05	0.04	0.01	0.02

Formula used:

  $\sigma =\sqrt{E\left(Y^2\right)+{\left\{E\left(Y\right)\right\}}^2}$

Calculation:

y	$\text{P}\left(y\right)$	$\text{yP}\left(y\right)$	$y^2$	$y^2\text{P}\left(y\right)$
0	0.68	0	0	0
1	0.05	0.05	1	0.05
2	0.07	0.14	4	0.28
3	0.08	0.24	9	0.72
4	0.05	0.2	16	0.8
5	0.04	0.2	25	1
6	0.01	0.06	36	0.36
7	0.02	0.14	49	0.98
		1.03		4.19

  $\begin{array}{c}E\left(Y\right)={\displaystyle \sum yP(y)}\\ =1.03\\ E\left(Y^2\right)=y^{2}P\left(y\right)\\ =4.19\end{array}$

  $\begin{array}{c}\sigma =\sqrt{4.19-{1.03}^2}\\ =\sqrt{3.1291}\\ =1.769\end{array}$

Standard deviation is 1.769 and mean is 1.03 therefore the days on an arbitrary choose varies 1.769 from the mean days 1.03