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5. Poisson counts
Arrivals, calls, defects: a rate per unit of time or space. Scale the rate to the window first.
| Count | P(exactly) | P(this many or fewer) |
|---|---|---|
| 0 | 0.0183 | 0.0183 |
| 1 | 0.0733 | 0.0916 |
| 2 | 0.1465 | 0.2381 |
| 3 | 0.1954 | 0.4335 |
| 4 | 0.1954 | 0.6288 |
| 5 | 0.1563 | 0.7851 |
| 6 | 0.1042 | 0.8893 |
| 7 | 0.0595 | 0.9489 |
| 8 | 0.0298 | 0.9786 |
| 9 | 0.0132 | 0.9919 |
| 10 | 0.0053 | 0.9972 |
- λ for this window44 per hour × 1 hour. Scale the rate first; this is the step people miss.
- P(exactly 3)0.1954e^−4 × 4^3 / 3!
- P(3 or fewer)0.4335Add P(0) through P(3)
- P(fewer than 3)0.2381Add P(0) through P(2)
- P(at least 3)0.76191 − P(2 or fewer)
- P(more than 3)0.56651 − P(3 or fewer)
- P(none)0.0183e^−4
- P(at least one)0.98171 − e^−4
- P(exactly 3 | at least one)0.199P(3) / P(at least one) = 0.1954 / 0.9817. Given at least one, divide by that.
- Mean and variance4Both equal λ for a Poisson. SD = √λ = 2