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Poisson Distribution

Poisson distribution calculator. Enter λ and k to compute P(X=k), mean and variance, with PMF notes—for count processes.

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P(X=k)=e^(−λ) λ^k / k!
average occurrences per interval
compute P(X = k)

📖 Tutorial | Poisson Distribution

1. Definition
Poisson distribution models the count X of rare events per interval: P(X=k)=e^(−λ)λ^k/k!, with mean λ; it is the limit of binomial for large n, small p.
2. Symbols
SymbolMeaning
λmean occurrences per interval
kactual count (non-negative integer)
E(X)=Var(X)=λmean and variance equal
3. How it works
  • Poisson pmf P(X=k)=e^(−λ)λ^k/k!;
  • Mean and variance both equal λ;
  • Requires λ>0 and k integer ≥0;
  • Used for call counts, defects, arrivals.
4. Steps
  1. Enter the rate λ, e.g. 3;
  2. Enter the count k, e.g. 2;
  3. Click Calculate for P(X=2);
  4. Read the mean and variance.
5. Example
Example: λ=3, find P(X=2). P=e^(−3)·3²/2!=0.0498×9/2≈0.2240.
6. Pitfalls
k must be a non-negative integer, not a decimal;
λ must be positive;
Poisson is an approximation: events must be independent and rare.

❓ FAQ | Poisson Distribution

Why are mean and variance equal?
A property of the Poisson pmf; both equal λ.
Relation to the binomial?
Binomial with large n, small p and np=λ is approx. Poisson.
Can k be a decimal?
No; k is a non-negative integer count.
What does λ mean?
The average occurrences per interval—the mean.
Where is it used?
Rare events per interval: calls per minute, typos, accidents.