Chapter 4.4, Problem 76E

(a)

To determine

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

Answer to Problem 76E

Solution: Variance and standard deviation of random variable Z are 2500 and 50, respectively.

Explanation of Solution

Calculation:

Consider X, Y, and Z to be the random variables with variances ${\sigma }_x^2,{\sigma }_y^2\text{, and }{\sigma }_z^2$. Consider a and b to be the constants, thatis,$a=2\text{ and }b=10$. 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:

$\begin{array}{c}{\sigma }_z^2=b^2{\sigma }_x^2\\ ={\left(10\right)}^2\times {\left(5\right)}^2\\ =2500\end{array}$

The standard deviation can be calculated as follows:

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

Substitute the values in the above formula:

$\begin{array}{c}{\sigma }_z=\sqrt{b^2{\sigma }_x^2}\\ =\sqrt{2500}\\ =50\end{array}$

The variance of the random variable Z is 2500and the standard deviation is 50.

(b)

To determine

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

Answer to Problem 76E

Solution: The variance and standard deviation of random variable Z are 2500 and 50, respectively.

Explanation of Solution

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

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

Substitute the values in the above formula:

$\begin{array}{c}{\sigma }_z^2=b^2{\sigma }_x^2\\ ={\left(10\right)}^2\times {\left(5\right)}^2\\ =2500\end{array}$

The standard deviation can be calculated as follows:

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

Substitute the values in the above formula:

$\begin{array}{c}{\sigma }_z^{}=\sqrt{b^2{\sigma }_x^2}\\ =\sqrt{2500}\\ =50\end{array}$

(c)

To determine

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

Answer to Problem 76E

Solution: The variance and standard deviation of random variable Z are 125 and 11.18, respectively.

Explanation of Solution

Calculation: Consider X, Y, and Z to be the random variables with 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:

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

Substitute the values in the above formula:

$\begin{array}{c}{\sigma }_z^2={\sigma }_x^2+{\sigma }_y^2\\ ={\left(5\right)}^2+{\left(10\right)}^2\\ =125\end{array}$

The standard deviation can be calculated as follows:

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

Substitute the values in the above formula:

$\begin{array}{c}{\sigma }_z^{}=\sqrt{{\sigma }_x^2+{\sigma }_y^2}\\ =\sqrt{125}\\ =11.18\end{array}$

(d)

To determine

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

Answer to Problem 76E

Solution: The variance and standard deviation of random variable Z are 125and 11.18, respectively.

Explanation of Solution

Calculation: Consider X, Y, and Z to be the random variables with 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:

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

Substitute the values in the above formula:

$\begin{array}{c}{\sigma }_z^2={\sigma }_x^2+{\sigma }_y^2\\ ={\left(5\right)}^2+{\left(10\right)}^2\\ =125\end{array}$

The standard deviation can be calculated as follows:

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

Substitute the values in the above formula:

$\begin{array}{c}{\sigma }_z^{}=\sqrt{{\sigma }_x^2+{\sigma }_y^2}\\ =\sqrt{125}\\ =11.18\end{array}$

(e)

To determine

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

Answer to Problem 76E

Solution: The variance and standard deviation of random variable Z are 625 and 25, respectively.

Explanation of Solution

Calculation: Consider X, Y, and Z to be the random variables with variances ${\sigma }_x^2,{\sigma }_y^2\text{ and }{\sigma }_z^2$. Consider a and b to be the constants, that is,$a=-3\text{ and }b=-2$. If $Z=-aX-bY$, then variance can be calculated as follows:

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

Substitute the values in the above formula:

$\begin{array}{c}{\sigma }_z^2=a^2{\sigma }_x^2+b^2{\sigma }_y^2\\ ={\left(-3\right)}^2\times {\left(5\right)}^2+{\left(-2\right)}^2\times {\left(10\right)}^2\\ =625\end{array}$

The standard deviation can be calculated as follows:

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

Substitute the values in the above formula:

$\begin{array}{c}{\sigma }_z^{}=\sqrt{625}\\ =25\end{array}$