Chapter 5.3, Problem 18E

The American Housing Survey reported the following data on the number of times that owner-occupied and renter-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months (U.S. Census Bureau website, October 2012).

	Number of Units (1000s)
Number of Times	Owner Occupied	Renter Occupied
0	439	394
1	1100	760
2	249	221
3	98	92
4 times or more	120	111

1. a. Define a random variable x = number of times that owner-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months and develop a probability distribution for the random variable. (Let x = 4 represent 4 or more times.)
2. b. Compute the expected value and variance for x.
3. c. Define a random variable y = number of times that renter-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months and develop a probability distribution for the random variable. (Let y = 4 represent 4 or more times.)
4. d. Compute the expected value and variance for y.
5. e. What observations can you make from a comparison of the number of water supply stoppages reported by owner-occupied units versus renter-occupied units?

To determine

Construct a discrete probability distribution for the random variable x.

Answer to Problem 18E

The probability distribution for the random variable x is given by,

x	f(x)
0	0.2188
1	0.5484
2	0.1241
3	0.0489
4	0.0598

Explanation of Solution

Calculation:

The data represents the number of times that owner-occupied and renter-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months. The random variable x is the number of times that owner-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months.

The corresponding probabilities of the random variable x are obtained by dividing the number of units that are owner occupied (f) with total number of units (N) that are owner occupied.

That is, $f\left(x\right)=\frac{f}{N}$

For example, $f\left(1\right)=0.2188\left(=\frac{439}{2006}\right)$

The probability distribution for the random variable x can be obtained as follows:

x	f	f(x)
0	439	0.2188
1	1100	0.5484
2	249	0.1241
3	98	0.0489
4	120	0.0598
Total	2600	1.0000

Thus, a discrete probability distribution for the random variable x is obtained.

To determine

Compute the expected value and variance for the random variable x.

Answer to Problem 18E

The expected value for the random variable x is 1.1825.

The variance of the random variable x is 1.0435.

Explanation of Solution

Calculation:

The formula for the expected value of a discrete random variable is,

$\begin{array}{c}E\left(x\right)=\mu \\ ={\displaystyle \sum x\cdot f\left(x\right)}\end{array}$

The formula for the variance of the discrete random variable is,

${\sigma }^2={\displaystyle \sum \left[{\left(x-\mu \right)}^2.f\left(x\right)\right]}$

The expected value and variance for the random variable x is obtained using the following table:

x	f(x)	$x\cdot f\left(x\right)$	$\left(x-\mu \right)$	${\left(x-\mu \right)}^2$	${\left(x-\mu \right)}^2.f\left(x\right)$
0	0.2188	0	–1.1825	1.398306	0.305949
1	0.5484	0.5484	–0.1825	0.033306	0.018265
2	0.1241	0.2482	0.8175	0.668306	0.082937
3	0.0489	0.1467	1.8175	3.303306	0.161532
4	0.0598	0.2392	2.8175	7.938306	0.474711
Total	1.0000	1.1825	4.0875	13.34153	1.043394

Thus, the expected value for the random variable x is 1.1825 and the variance of the random variable x is 1.0435.

To determine

Construct a discrete probability distribution for the random variable y.

Answer to Problem 18E

The probability distribution for the random variable y is given by,

y	f(y)
0	0.2188
1	0.5484
2	0.1241
3	0.0489
4	0.0598

Explanation of Solution

Calculation:

The data represents the number of times that owner-occupied and renter-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months. The random variable y is the number of times that renter-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months.

The corresponding probabilities of the random variable y are obtained by dividing the number of units that are owner occupied (f) with total number of units (N) that are owner occupied.

That is, $f\left(y\right)=\frac{f}{N}$

For example, $f\left(1\right)=0.2188\left(=\frac{394}{1,578}\right)$

The probability distribution for the random variable y can be obtained as follows:

y	f	f(y)
0	394	0.249683
1	760	0.481622
2	221	0.140051
3	92	0.058302
4	111	0.070342
Total	1,578	1

Thus, a discrete probability distribution for the random variable y is obtained.

To determine

Compute the expected value and variance for the random variable y.

Answer to Problem 18E

The expected value for the random variable y is 1.2180.

The variance of the random variable y is 1.2085.

Explanation of Solution

Calculation:

The formula for the expected value of a discrete random variable is,

$\begin{array}{c}E\left(y\right)=\mu \\ ={\displaystyle \sum y\cdot f\left(y\right)}\end{array}$

The formula for the variance of the discrete random variable is,

${\sigma }^2={\displaystyle \sum \left[{\left(y-\mu \right)}^2.f\left(y\right)\right]}$

The expected value and variance for the random variable y is obtained using the following table:

y	f(y)	$y\cdot f\left(y\right)$	$\left(y-\mu \right)$	${\left(y-\mu \right)}^2$	${\left(y-\mu \right)}^2.f\left(y\right)$
0	0.249683	0	–1.218	1.483518	0.370409
1	0.481622	0.481622	–0.218	0.047523	0.022888
2	0.140051	0.280101	0.782003	0.611528	0.085645
3	0.058302	0.174905	1.782003	3.175533	0.185139
4	0.070342	0.281369	2.782003	7.739538	0.544416
Total	1	1.217997	3.910013	13.05764	1.208497

Thus, the expected value for the random variable y is 1.2180 and the variance of the random variable y is 1.2085.

To determine

Explain the observations that can be made from a comparison of the number of water supply stoppages reported by owner-occupied units versus renter-occupied units.

Explanation of Solution

The expected number of times that renter-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months is 1.2180 and the expected number of times that owner-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months is 1.1825. Thus, the expected value is greater for the renter-occupied units than the owner-occupied units. The variability is less for the owner-occupied units comparing to the renter-occupied units.