Chapter PIV, Problem 14RE

To determine

To find: the standard deviation and mean of the number of spots might move.

Answer to Problem 14RE

Mean = 35.83

Standard Deviation = 6.31

Explanation of Solution

Formula used:

  $\begin{array}{c}E\left(Z\right)=E\left(X-Y\right)\\ =E\left(X\right)-E\left(Y\right)\\ Var\left(Z\right)=\mathrm{var}\left(X-Y\right)\\ Var\left(Z\right)=\mathrm{var}\left(X+Y\right)\\ =\mathrm{var}\left(X\right)+\mathrm{var}\left(Y\right)\\ SD\left(Z\right)=\sqrt{\mathrm{var}\left(Z\right)}\end{array}$

Calculation:

Suppose Z is the move when in "Penalty Zone" it land.

Z = X −Y

Mean of Z is

  $\begin{array}{c}E\left(Z\right)=E\left(X-Y\right)\\ =E\left(X\right)-E\left(Y\right)\\ =37.5-\frac{10}{6}\\ =35.83\end{array}$

Variance:

  $\begin{array}{c}Var\left(Z\right)=\mathrm{var}\left(X-Y\right)\\ Var\left(Z\right)=\mathrm{var}\left(X+Y\right)\\ =\mathrm{var}\left(X\right)+\mathrm{var}\left(Y\right)\\ =37.5+2.33\\ =39.83\end{array}$

Standard deviation:

  $\begin{array}{c}SD\left(Z\right)=\sqrt{\mathrm{var}\left(Z\right)}\\ =\sqrt{39.83}\\ =6.31\end{array}$