An exhaustive event set contains events whose union covers the entire sample space. Consequently, at least one event in the set must occur. When the events are also mutually exclusive, their probabilities sum to 1; in continuous domains, the corresponding probability density integrates to 1. section 5.10 section 5.6
Use exhaustiveness to check whether a collection accounts for every possible outcome:
- For discrete outcomes, the union of the events must include every allowed value of the random variable. section 5.2
- For continuous outcomes, infinitesimal intervals across the complete domain form an exhaustive set; summation becomes integration. section 5.6
- Exhaustiveness guarantees coverage, while mutual exclusivity guarantees that an outcome is not counted in multiple events. section 5.2
Quantized adult weights
Partition adult weights into S1 (<60 kg), S2 (60–90 kg), and S3 (>90 kg). These ranges are exhaustive because every weight belongs to one of them. They are mutually exclusive, and their probabilities—0.25, 0.50, and 0.25—sum to 1. section 5.2
Table 5.2 Probability distribution for quantized weights of Statsville adults
| S1: Less than 60 kg |
S2: Between 60 and 90 kg |
S3: More than 90 kg |
|---|---|---|
| p(X = 1) = 25,000/100,000 = 0.25 | p(X = 2) = 50,000/100,000 = 0.5 | p(X = 3) = 25,000/100,000 = 0.25 |
Inspect the partition and its probabilities as a compact table. section 5.2
