Explanation 1
Exact model: The original fixed-trial binomial probability.
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Lesson 13 of 18
Compare exact binomial probabilities with a labelled Poisson approximation using lambda=np.
Advanced / Optional
Compare exact binomial probabilities with a labelled Poisson approximation using lambda=np.
Explanation 1
Exact model: The original fixed-trial binomial probability.
Explanation 2
Approximation: A Poisson value used when events are rare and n is large.
Explanation 3
Error: Difference between exact and approximate values.
Terminology
The original fixed-trial binomial probability.
dbinom result.
A Poisson value used when events are rare and n is large.
dpois with lambda=np.
Difference between exact and approximate values.
Absolute or relative error.
Notation and formulas
Binomial(n,p) approximately Poisson(lambda=np)
Define X, its support, parameters, units, and assumptions before substituting values.
Interactive Mission
Compare an exact binomial PMF with a labelled approximation.
P(X = x)
0.2182
F(x) = P(X <= x)
0.6296
E(X)
4
SD(X)
1.7889
| x | P(X=x) | Included in F(4) |
|---|---|---|
| 0 | 0.0115 | Yes |
| 1 | 0.0576 | Yes |
| 2 | 0.1369 | Yes |
| 3 | 0.2054 | Yes |
| 4 | 0.2182 | Yes |
| 5 | 0.1746 | No |
| 6 | 0.1091 | No |
| 7 | 0.0545 | No |
| 8 | 0.0222 | No |
| 9 | 0.0074 | No |
| 10 | 0.002 | No |
| 11 | 0.0005 | No |
| 12 | 0.0001 | No |
| 13 | 0 | No |
| 14 | 0 | No |
| 15 | 0 | No |
| 16 | 0 | No |
| 17 | 0 | No |
| 18 | 0 | No |
| 19 | 0 | No |
| 20 | 0 | No |
Interpretation: Custom PMF, discrete uniform, Bernoulli, or binomial: inspect support and trial structure before choosing.
Approximation safeguard: Treat the comparison model as approximate and inspect its error against the exact model.
Worked example
Scenario
A large set of tickets has a small stable escalation probability.
Approximation quality varies by parameters and query.
R connection
Run and edit the original script, inspect intermediate output, and keep exact and approximate results clearly labelled.
Live R Lab
Compare exact binomial and approximate Poisson probabilities.
Ready to run
Dataset provenance
Original synthetic data generated for STATLAB Academy. No textbook data used.
Interpretation safeguard
Never present the Poisson approximation as exact; compare with the exact binomial whenever feasible.
Reflection
Name the random variable, support, parameters, units, dependence assumptions, and one limitation.
Assistant
Coach me through Poisson Approximation to the Binomial 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
Explore additional discrete-distribution explanations from properly licensed or publisher resources.
DownloadResource
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