Chapter 6, Problem 2RE

a.

To determine

To find the mean of the given probability distribution.

Answer to Problem 2RE

Mean = 8.37

Explanation of Solution

Given:

x	P ( x )
6	0.21
7	0.12
8	0.29
9	0.11
10	0.01
11	0.26

Concept used:

  $\text{Mean=}\sum x.P\left(x\right)$

x	P ( x )	$x.P\left(x\right)$
6	0.21	1.26
7	0.12	0.84
8	0.29	2.32
9	0.11	0.99
10	0.01	0.1
11	0.26	2.86
Sum	1	8.37

Since the $\sum x.P\left(x\right)$ = 8.37, from the above table.

Therefore, the mean of the given distribution is 8.37.

b.

To determine

To find the Variance of the given probability distribution.

Answer to Problem 2RE

Variance = 3.3131

Explanation of Solution

Given:

x	P ( x )
6	0.21
7	0.12
8	0.29
9	0.11
10	0.01
11	0.26

Concept used:

  $\text{Variance=}\sum x^2.P\left(x\right)-{\left[\sum x.P\left(x\right)\right]}^2$

x	P ( x )	$x.P\left(x\right)$	$x^2.P\left(x\right)$
6	0.21	1.26	7.56
7	0.12	0.84	5.88
8	0.29	2.32	18.56
9	0.11	0.99	8.91
10	0.01	0.1	1
11	0.26	2.86	31.46
Sum	1	8.37	73.37

From the above table:

  $\begin{array}{l}\sum x.P\left(x\right)=8.37\\ \sum x^2.P\left(x\right)=73.37\end{array}$

Therefore, the variance is,

  $\begin{array}{c}\text{Variance=}\sum x^2.P\left(x\right)-{\left[\sum x.P\left(x\right)\right]}^2\\ =73.37-{\left(8.37\right)}^2\\ =3.3131\end{array}$

c.

To determine

To find the Standard Deviation of the given probability distribution.

Answer to Problem 2RE

Standard Deviation = 1.82

Explanation of Solution

Given:

x	P ( x )
6	0.21
7	0.12
8	0.29
9	0.11
10	0.01
11	0.26

Concept used:

  $\text{Standard Deviation =}\sqrt{\sum x^2.P\left(x\right)-{\left[\sum x.P\left(x\right)\right]}^2}$

x	P ( x )	$x.P\left(x\right)$	$x^2.P\left(x\right)$
6	0.21	1.26	7.56
7	0.12	0.84	5.88
8	0.29	2.32	18.56
9	0.11	0.99	8.91
10	0.01	0.1	1
11	0.26	2.86	31.46
Sum	1	8.37	73.37

From the above table:

  $\begin{array}{l}\sum x.P\left(x\right)=8.37\\ \sum x^2.P\left(x\right)=73.37\end{array}$

Therefore, the variance is,

  $\begin{array}{c}\text{Standard Deviation=}\sqrt{\sum x^2.P\left(x\right)-{\left[\sum x.P\left(x\right)\right]}^2}\\ =\sqrt{73.37-{\left(8.37\right)}^2}\\ =\sqrt{3.3131}\\ =1.82\end{array}$