Poisson Distribution Calculator

Poisson Distribution Formula

Worked Example

Exactly 3 events when 2 are expected (λ = 2):

1. P(X = k) = e^(−λ) λᵏ ÷ k!
2. = e^(−2) × 2^3 ÷ 3!
3. = 0.18044704

P(X < 3): 0.67667642 · P(X ≥ 3): 0.32332358 · P(X > 3): 0.14287654 · Mean = variance = λ: 2 · Standard deviation √λ: 1.41421

Scaling λ to the interval

λ must be the expected number of events in the interval you ask about. If calls arrive at 6 per hour, the number in 20 minutes has λ = 2. The Poisson model assumes events are independent and occur at a constant rate; bunching or a changing rate makes it inaccurate. It also approximates the binomial distribution when n is large and p small, with λ = np.

More Engineering Mathematics Calculators

• Binomial Distribution Calculator

  P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ

  n trials, p probability of success, mean np, variance np(1 − p)

  Example: 3 heads in 10 tosses of a fair coin: P(X = 3) = 0.1171875.

  Open the binomial distribution calculator →

• Eigenvalue Calculator

  det(A − λI) = 0

  Σλ = trace A, Πλ = det A

  Example: The matrix [[4, 1], [2, 3]]: Eigenvalues: λ₁ = 5, λ₂ = 2.

  Open the eigenvalue calculator →

• Simpson’s Rule Calculator

  I = h/3 [y₀ + 4(y₁ + y₃ + …) + 2(y₂ + y₄ + …) + yₙ]

  h step size, n even number of intervals

  Example: y = 1/(1 + x²) at x = 0, 0.25, …, 1 (h = 0.25): Integral I ≈ 0.785392.

  Open the simpson’s rule calculator →

Frequently asked questions

Sources

Last reviewed: 2026-10-15. Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.