Chapter PIV, Problem 23RE

(a)

To determine

To Calculate: the mean and standard deviation.

Answer to Problem 23RE

Mean =100

Standard Deviation = 8

Explanation of Solution

Given:

Provide table

	Mean	SD
X	50	8
Y	100	6

Calculation:

Mean:

  $\begin{array}{c}E\left(X+50\right)=E\left(X\right)+50\\ =50+50\\ =100\end{array}$

Variance:

  $\begin{array}{c}\mathrm{var}\left(X+50\right)=\mathrm{var}\left(X\right)\\ =8^2\\ =64\end{array}$

Standard deviation:

  $\begin{array}{c}SD\left(X+50\right)=\sqrt{\mathrm{var}\left(X+50\right)}\\ =\sqrt{64}\\ =8\end{array}$

(b)

To determine

To Calculate: the mean and standard deviation.

Answer to Problem 23RE

Mean =1000

Standard Deviation = 60

Explanation of Solution

Given:

Provide table

	Mean	SD
X	50	8
Y	100	6

Calculation:

Mean:

  $\begin{array}{c}E\left(10Y\right)=10E\left(Y\right)\\ =10\left(100\right)\\ =1000\end{array}$

Variance:

  $\begin{array}{c}\mathrm{var}\left(10y\right)={10}^2\mathrm{var}\left(Y\right)\\ =100{\left(6\right)}^2\\ =3600\end{array}$

Standard deviation:

  $\begin{array}{c}SD\left(10Y\right)=\sqrt{3600}\\ =60\end{array}$

(c)

To determine

To Calculate: the mean and standard deviation.

Answer to Problem 23RE

Mean =10

Standard Deviation = 8.54

Explanation of Solution

Given:

Provide table

	Mean	SD
X	50	8
Y	100	6

Calculation:

Mean:

  $\begin{array}{c}E\left(X+0.5Y\right)=E\left(X\right)+0.5E\left(Y\right)\\ =50+0.5\left(100\right)\\ =100\end{array}$

Variance:

  $\begin{array}{c}\mathrm{var}\left(X+0.5Y\right)=\mathrm{var}\left(X\right)+{\left(0.5\right)}^2\mathrm{var}\left(y\right)\\ =8^2+0.25{\left(6\right)}^2\\ =64+0.25\left(36\right)\\ =64+9\\ =73\end{array}$

Standard deviation:

  $\begin{array}{c}SD\left(X+0.5Y\right)=\sqrt{\mathrm{var}\left(X+0.5Y\right)}\\ =\sqrt{73}\\ =8.54\end{array}$

(d)

To determine

To Calculate: the mean and standard deviation.

Answer to Problem 23RE

Mean =-50

Standard Deviation = 10

Explanation of Solution

Given:

Provide table

	Mean	SD
X	50	8
Y	100	6

Calculation:

Mean:

  $\begin{array}{c}E\left(X-Y\right)=E\left(X\right)-E\left(Y\right)\\ =50-100\\ =-50\end{array}$

Variance:

  $\begin{array}{c}\mathrm{var}\left(X-Y\right)=\mathrm{var}\left(X\right)+\mathrm{var}\left(Y\right)\\ =8^2+6^2\\ =64+36\\ =100\end{array}$

Standard deviation:

  $\begin{array}{c}SD\left(X-Y\right)=\sqrt{\mathrm{var}\left(X-Y\right)}\\ =\sqrt{100}\\ =10\end{array}$

(e)

To determine

To Calculate: the mean and standard deviation.

Answer to Problem 23RE

Mean =100

Standard Deviation = 11.3

Explanation of Solution

Given:

Provide table

	Mean	SD
X	50	8
Y	100	6

Calculation:

Mean:

  $\begin{array}{c}E\left(X_1+X_2\right)=E\left(X_1\right)+E\left(X_2\right)\\ =50+50\\ =100\end{array}$

Variance:

  $\begin{array}{c}\mathrm{var}\left(X_1+X_2\right)=\mathrm{var}\left(X_1\right)+\mathrm{var}\left(X_2\right)\\ =8^2+8^2\\ =128\end{array}$

Standard deviation:

  $\begin{array}{c}SD\left(X_1+X_2\right)=\sqrt{\mathrm{var}\left(X_1+X_2\right)}\\ =\sqrt{128}\\ =11.3\end{array}$