Chapter 6.1, Problem 16E

To determine

To find: The standard deviation of the random variable X and interpret it

Answer to Problem 16E

The standard deviation is 1.3106.

Explanation of Solution

Given:

The probability distribution is:

Value of X	0	1	2	3	4	5
Probability	0.03	0.16	0.30	0.23	0.17	0.11

Formula used:

The formula to compute the mean is:

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

Calculation:

The mean can be calculated as:

  $\begin{array}{c}\text{Mean =}{\displaystyle \sum x\times P\left(x\right)}\\ =0\left(0.03\right)+1\left(0.16\right)+2\left(0.30\right)+3\left(0.23\right)+4\left(0.17\right)+5\left(0.11\right)\\ =2.68\end{array}$

The standard deviation is calculated as follows:

  $\begin{array}{c}\sigma \text{=}\sqrt{{\displaystyle \sum x^2\times P\left(x\right)}-{\left({\displaystyle \sum x\times P\left(x\right)}\right)}^2}\\ =\sqrt{0^2\times 0.03+1^2\times 0.16+\mathrm{....}+5^2\times 0.11-{\left(2.68\right)}^2}\\ =1.306\end{array}$

Thus, the standard deviation is 1.3106.

Interpretation:

The number of toys played are 1.3106 on average away from the mean.