The following data has been collected 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. Number of Times Number of Units (1,000s) Owner Occupied Renter Occupied 0 439 394 1 1,200 760 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. Round your answers to four decimal places.) x f(x) xf(x) x−μ (x − μ)2 (x − μ)2f(x) 0 1 2 3 4 (b) Compute the expected value and variance for x. (Round your answers to four decimal places.) expected value variance
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
| Number of Times | Number of Units (1,000s) | |
|---|---|---|
| Owner Occupied | Renter Occupied | |
| 0 | 439 | 394 |
| 1 | 1,200 | 760 |
| 2 | 249 | 221 |
| 3 | 98 | 92 |
| 4 times or more | 120 | 111 |
| x |
f(x)
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xf(x)
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x−μ
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(x − μ)2
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(x − μ)2f(x)
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|---|---|---|---|---|---|
| 0 | |||||
| 1 | |||||
| 2 | |||||
| 3 | |||||
| 4 |
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