Chapter 4.4, Problem 73E

(a)

To determine

To find: The variance and standard deviation of the random variable Z.

Answer to Problem 73E

Solution: The variance and standard deviation of random variable Z are 1600 and 40, respectively.

Explanation of Solution

Calculation: Consider X, Y, and Z to be the random variables with averages ${\mu }_x,{\mu }_y\text{, and }{\mu }_z$ and variances ${\sigma }_x^2,{\sigma }_y^2\text{, and }{\sigma }_z^2$. Consider a and b to be the constants, that is, $a=35\text{ and }b=-10$. If $Z=a-bX$, then variance can be calculated as follows:

${\sigma }_z^2=b^2{\sigma }_x^2$

Substitute the values in the above formula:

$\begin{array}{c}{\sigma }_z^2=b^2{\sigma }_x^2\\ ={\left(-10\right)}^2\times {\left(4\right)}^2\\ =1600\end{array}$

The standard deviation can be calculated as follows:

${\sigma }_z^{}=\sqrt{b^2{\sigma }_x^2}$

Substitute the values in the above formula:

$\begin{array}{c}{\sigma }_z=\sqrt{b^2{\sigma }_x^2}\\ =\sqrt{1600}\\ =40\end{array}$

(b)

To determine

To find: The variance and standard deviation of the random variable Z.

Answer to Problem 73E

Solution: The variance and standard deviation of random variable Z are 2304 and 48, respectively.

Explanation of Solution

Calculation: Consider X, Y, and Z to be the random variables with averages ${\mu }_x,{\mu }_y\text{, and }{\mu }_z$ and variances ${\sigma }_x^2,{\sigma }_y^2\text{, and }{\sigma }_z^2$. Consider a and b to be the constants, that is, $a=-5\text{ and }b=12$. If $Z=a-bX$, then variance can be calculated as follows:

${\sigma }_z^2=b^2{\sigma }_x^2$

Substitute the values in the above formula:

$\begin{array}{c}{\sigma }_z^2=b^2{\sigma }_x^2\\ ={\left(12\right)}^2\times {\left(4\right)}^2\\ =2304\end{array}$

The standard deviation can be calculated as follows:

${\sigma }_z=\sqrt{b^2{\sigma }_x^2}$

Substitute the values in the above formula:

$\begin{array}{c}{\sigma }_z^{}=\sqrt{b^2{\sigma }_x^2}\\ =\sqrt{2304}\\ =48\end{array}$

(c)

To determine

To find: The variance and standard deviation of the random variable Z.

Answer to Problem 73E

Solution: The variance and standard deviation of random variable Z are 112 and 10.6, respectively.

Explanation of Solution

Calculation: Consider X, Y, and Z to be the random variables with averages ${\mu }_x,{\mu }_y\text{, and }{\mu }_z$ and variances ${\sigma }_x^2,{\sigma }_y^2\text{, and }{\sigma }_z^2$. Consider a and b to be the constants, that is, $a=1\text{ and }b=1$. If $Z=aX+aY$, then variance can be calculated as follows:

${\sigma }_z^2={\sigma }_x^2+{\sigma }_y^2+2\rho {\sigma }_x{\sigma }_y$

Substitute the values in the above formula:

$\begin{array}{c}{\sigma }_z^2={\sigma }_x^2+{\sigma }_y^2+2\rho {\sigma }_x{\sigma }_y\\ ={\left(4\right)}^2+{\left(8\right)}^2+2\times 0.5\times 4\times 8\\ =112\end{array}$

The standard deviation can be calculated as follows:

${\sigma }_z^{}=\sqrt{{\sigma }_x^2+{\sigma }_y^2+2\rho {\sigma }_x{\sigma }_y}$

Substitute the values in the above formula:

$\begin{array}{c}{\sigma }_z^{}=\sqrt{{\sigma }_x^2+{\sigma }_y^2+2\rho {\sigma }_x{\sigma }_y}\\ =\sqrt{112}\\ =10.6\end{array}$

(d)

To determine

To find: The variance and standard deviation of the random variable Z.

Answer to Problem 73E

Solution: The variance and standard deviation of random variable Z are 48 and 6.9, respectively.

Explanation of Solution

Calculation: Consider X, Y, and Z to be the random variables with averages ${\mu }_x,{\mu }_y\text{, and }{\mu }_z$ and variances ${\sigma }_x^2,{\sigma }_y^2\text{, and }{\sigma }_z^2$. Consider a and b to be the constants, that is, $a=1\text{ and }b=-1$. If $Z=aX-bY$, then variance can be calculated as follows:

${\sigma }_z^2={\sigma }_x^2+{\sigma }_y^2-2\rho {\sigma }_x{\sigma }_y$

Substitute the values in the above formula:

$\begin{array}{c}{\sigma }_z^2={\sigma }_x^2+{\sigma }_y^2-2\rho {\sigma }_x{\sigma }_y\\ ={\left(4\right)}^2+{\left(8\right)}^2-2\times 0.5\times 4\times 8\\ =48\end{array}$

The standard deviation can be calculated as follows:

${\sigma }_z^{}=\sqrt{{\sigma }_x^2+{\sigma }_y^2-2\rho {\sigma }_x{\sigma }_y}$

Substitute the values in the above formula:

$\begin{array}{c}{\sigma }_z^{}=\sqrt{{\sigma }_x^2+{\sigma }_y^2-2\rho {\sigma }_x{\sigma }_y}\\ =\sqrt{48}\\ =6.9\end{array}$

(e)

To determine

To find: The variance and standard deviation of the random variable Z.

Answer to Problem 73E

Solution: The variance and standard deviation of random variable Z are 192 and 13.85, respectively.

Explanation of Solution

Calculation: Correlation can be calculated as follows:

$r=\frac{Cov\left(x,y\right)}{{\sigma }_x{\sigma }_y}$

Substitute the provided values in the above formula:

$\begin{array}{c}r=\frac{Cov\left(x,y\right)}{{\sigma }_x{\sigma }_y}\\ Cov\left(x,y\right)=2\times 0.5\times \left(-2\right)\times 2\times 4\times 8\\ =-128\end{array}$

Consider X, Y, and Z to be the random variables with averages ${\mu }_x,{\mu }_y\text{, and }{\mu }_z$ and variances ${\sigma }_x^2,{\sigma }_y^2\text{, and }{\sigma }_z^2$. Consider a and b to be the constants, that is, $a=-2\text{ and }b=2$. If $Z=aX+bY$, then variance can be calculated as follows:

${\sigma }_z^2=a^2{\sigma }_x^2+b^2{\sigma }_y^2+2Cov\left(-2x,2y\right)$

Substitute the values in the above formula:

$\begin{array}{c}{\sigma }_z^2=a^2{\sigma }_x^2+b^2{\sigma }_y^2+2Cov\left(-2x,2y\right)\\ ={\left(-2\right)}^2\times {\left(4\right)}^2+{\left(2\right)}^2\times {\left(8\right)}^2+2\times 0.5\times \left(-2\right)\times 2\times 4\times 8\\ =192\end{array}$

The standard deviation can be calculated as follows:

${\sigma }_z^{}=\sqrt{a^2{\sigma }_x^2+b^2{\sigma }_y^2+2Cov\left(-2x,2y\right)}$

Substitute the values in the above formula:

$\begin{array}{c}{\sigma }_z^{}=\sqrt{192}\\ =13.85\end{array}$