Concept explainers
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 |
- 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.)
- b. Compute the expected value and variance for x.
- 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.)
- d. Compute the expected value and variance for y.
- 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?
a.
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,
For example,
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.
b.
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,
The formula for the variance of the discrete random variable is,
The expected value and variance for the random variable x is obtained using the following table:
x | f(x) | |||||
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.
c.
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,
For example,
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.
d.
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,
The formula for the variance of the discrete random variable is,
The expected value and variance for the random variable y is obtained using the following table:
y | f(y) | |||||
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.
e.
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.
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