8.4 Continuous Distributions
- Key Idea 1: Saying that the data are continuous simply means that one cannot list all the possible values because, in principle, there are infinitely many...
Discussion
- For example, a lightbulb's lifetime (the length of time it is continuously on before breaking) cannot be less than 0 hours,although if it is defective it may last exactly 0 hours. There is some maximum length of time-if not 1000 hours, then maybe 10,000 hours or more. But any value between 0 and some large maximum is possible.
- Of course, in practice one cannot find the exact lifetime, but instead finds it to the nearest minute, second, or maybe 10th of a second.
- In any case, the actual number of possible values is immense; theoretically, it is infinite if one ignores the fact that all measurements are rounded off.
Key Idea 2: Here are some contrasting examples of continuous and discrete situations...
Key Idea 3: How does one specify probabilities for a continuous distribution?
Discussion
- With an infinite number of possible values, the probability of any particular value is 0.
- Recall the continuous chi-square distribution. We find a probability by computing an area in an interval (such as all numbers >= 2.8) under the chi-square curve.
- In general, we compute a probability by computing the area under a density curve within a given interval.
Example