Explanation 1
Point mass: dpois gives exactly x events.
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Lesson 12 of 18
Calculate Poisson point and tail probabilities while matching lambda to the requested exposure.
Core lesson
Calculate Poisson point and tail probabilities while matching lambda to the requested exposure.
Explanation 1
Point mass: dpois gives exactly x events.
Explanation 2
Cumulative mass: ppois gives at most x events.
Explanation 3
Rate scaling: Multiply lambda by the exposure ratio.
Terminology
dpois gives exactly x events.
Exactly ten calls.
ppois gives at most x events.
At most ten calls.
Multiply lambda by the exposure ratio.
30-minute to 90-minute rate.
Notation and formulas
P(X=x)=exp(-lambda)lambda^x/x!
Define X, its support, parameters, units, and assumptions before substituting values.
Interactive Mission
Inspect the event-count PMF for a selected rate.
P(X = x)
0.1339
F(x) = P(X <= x)
0.2851
E(X)
5.9916
SD(X)
2.4379
| x | P(X=x) | Included in F(4) |
|---|---|---|
| 0 | 0.0025 | Yes |
| 1 | 0.0149 | Yes |
| 2 | 0.0446 | Yes |
| 3 | 0.0892 | Yes |
| 4 | 0.1339 | Yes |
| 5 | 0.1606 | No |
| 6 | 0.1606 | No |
| 7 | 0.1377 | No |
| 8 | 0.1033 | No |
| 9 | 0.0688 | No |
| 10 | 0.0413 | No |
| 11 | 0.0225 | No |
| 12 | 0.0113 | No |
| 13 | 0.0052 | No |
| 14 | 0.0022 | No |
| 15 | 0.0009 | No |
Interpretation: Custom PMF, discrete uniform, Bernoulli, or binomial: inspect support and trial structure before choosing.
Interactive Mission
Scale lambda only after matching the exposure units.
P(X = x)
0.1339
F(x) = P(X <= x)
0.2851
E(X)
5.9916
SD(X)
2.4379
| x | P(X=x) | Included in F(4) |
|---|---|---|
| 0 | 0.0025 | Yes |
| 1 | 0.0149 | Yes |
| 2 | 0.0446 | Yes |
| 3 | 0.0892 | Yes |
| 4 | 0.1339 | Yes |
| 5 | 0.1606 | No |
| 6 | 0.1606 | No |
| 7 | 0.1377 | No |
| 8 | 0.1033 | No |
| 9 | 0.0688 | No |
| 10 | 0.0413 | No |
| 11 | 0.0225 | No |
| 12 | 0.0113 | No |
| 13 | 0.0052 | No |
| 14 | 0.0022 | No |
| 15 | 0.0009 | No |
Interpretation: Custom PMF, discrete uniform, Bernoulli, or binomial: inspect support and trial structure before choosing.
Worked example
Scenario
A support desk plans for an hourly interval from a 15-minute rate.
A unit mismatch can be four times more consequential than rounding.
R connection
Run and edit the original script, inspect intermediate output, and keep exact and approximate results clearly labelled.
Live R Lab
Model arrivals, scale rates to matching units, and compare observed mean and variance.
Ready to run
Dataset provenance
Original synthetic data generated for STATLAB Academy. No textbook data used.
Interpretation safeguard
State lambda with its exposure units every time it is used or scaled.
Reflection
Name the random variable, support, parameters, units, dependence assumptions, and one limitation.
Assistant
Coach me through Poisson Probabilities and Rate Scaling by asking about X, support, parameters, assumptions, probability notation, exact versus approximate status, and interpretation. Do not provide a quiz answer before submission or write my final capstone recommendation.
Exit check
Use conceptual hints before submitting. Full formulas, calculations, and interpretation appear afterward.
Checkpoint
Question 1 of 2. Answered 0/2. Passing score: 70%.
Resource
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