How to read the table: each cell shows P(X = k) = e−λ·λk/k! Rows: values of k. Columns: values of λ. For k much larger than λ the probabilities are practically 0 (shown as 0.0000).
Click a cell to read the value.
P(X = k) for λ from 0.5 to 10
How to read the table: each cell shows P(X ≤ k) = Σᵢ₌₀ᵏ e−λ·λi/i! For P(X > k) use the complement: 1 − P(X ≤ k). For P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a − 1).
Click a cell to read the value.
P(X ≤ k) for λ from 0.5 to 10
How to use these tables
Poisson Distribution
The Poisson distribution models the number of events occurring in an interval of time or space, when the events occur independently at a mean rate λ. Its probability mass function is:
\( P(X = k) = \dfrac{e^{-\lambda}\,\lambda^k}{k!}, \quad k = 0, 1, 2, \ldots \)
The mean and variance coincide: E[X] = Var[X] = λ.
Useful properties
- Sum of Poissons: if X ~ Po(λ₁) and Y ~ Po(λ₂) are independent, then X + Y ~ Po(λ₁ + λ₂).
- Approximation to the binomial: B(n, p) ≈ Po(np) when n is large and p is small (np ≤ 5 as a guideline).
- Normal approximation: for large λ, Po(λ) ≈ N(λ, λ).