Poisson Distribution Calculators

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The Poisson distribution models the probability of a given number of events occurring in a fixed interval of time or space, when events occur independently at a constant average rate (λ). P(k) = (λᵏ × e⁻λ) / k!, where k = number of events and λ = expected number of events (the mean). Key property: mean = variance = λ. The Poisson distribution is appropriate for count data representing rare events — radioactive decay counts, bacterial colony counts, mutations per genome per generation, cell arrivals at receptors, phone calls to a call center, traffic accidents per week. It approximates the binomial distribution when n is large and p is small (np = λ).

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Poisson Formula

P(k) = (λᵏ × e⁻λ) / k!

λ = mean number of events; k = 0, 1, 2, ... (non-negative integer); e ≈ 2.71828. Mean = λ; Variance = λ; SD = √λ.

Example: average 3 bacteria per mL (λ = 3). P(k=0) = (3⁰ × e⁻³) / 0! = e⁻³ = 0.0498. P(k=2) = (3² × e⁻³) / 2! = 9 × 0.0498 / 2 = 0.224 (22.4% chance of exactly 2 bacteria). P(k≤2) = P(0) + P(1) + P(2) = 0.0498 + 0.149 + 0.224 = 0.423.

When to Use Poisson Distribution

  • Events occur independently (one event doesn't affect probability of another)
  • Events occur at constant average rate (λ) over time or space
  • Events are rare relative to the number of opportunities

Biological Applications

Viral/bacterial counting: expected number of cells per hemocytometer square; CFU/mL calculations. Radioactive decay: disintegrations per minute follow Poisson. Mutation rate: mutations per genome per replication cycle. Colony picking: probability of picking N colonies from a plate to represent diversity.

Glossary

Poisson Distribution
P(k) = λᵏe⁻λ/k!; models count of independent rare events; mean = variance = λ; used for bacterial counts, radioactive decay, mutations per genome, MOI calculations.
λ (Lambda)
The Poisson distribution mean and rate parameter; equals both the mean and variance of the distribution; represents the expected number of events in the defined interval.
MOI (Multiplicity of Infection)
Average number of virus particles per cell when infecting a cell culture; MOI = 3: Poisson predicts e⁻³ ≈ 5% of cells uninfected; used to calculate infection efficiency in virology experiments.

Frequently Asked Questions

The Poisson distribution models the number of independent events occurring in a fixed time or space interval when the average rate λ is constant. Formula: P(k) = λᵏe⁻λ/k!. Parameters: λ = mean and variance = SD² (mean = variance is the key property). Used when: events occur independently; constant average rate; rare events relative to the number of opportunities (approximation of binomial with large n and small p). Examples: Counts of bacteria per microscope field. Radioactive decay events per minute. Number of mutations per genome per generation. Number of accidents per week on a stretch of road. Traffic arrivals per second at a server.

P(k) = λᵏ × e⁻λ / k!. Example: Poisson λ = 2 (average 2 mutations per gene). P(0 mutations) = 2⁰ × e⁻² / 0! = 1 × 0.1353 / 1 = 0.1353 (13.5% of genes have no mutations). P(1 mutation) = 2¹ × e⁻² / 1! = 2 × 0.1353 = 0.2707 (27.1%). P(2 mutations) = 2² × e⁻² / 2! = 4 × 0.1353 / 2 = 0.2707. P(3) = 2³ × e⁻² / 6 = 0.1804. Cumulative: P(k ≤ 2) = 0.1353 + 0.2707 + 0.2707 = 0.6767. Table or software (ppois in R; poisson.pmf in Python/scipy) for fast calculation.

The Poisson distribution is a limiting case of the binomial distribution when n is large and p is small (rare events): Binomial: P(k) = C(n,k) × pᵏ × (1−p)^(n−k). As n → ∞ and p → 0 with np = λ constant: P(k) → λᵏe⁻λ/k! (Poisson). Rule of thumb: Poisson approximation is good when n > 50 and p < 0.05 (or equivalently np = λ < 5). Example: monitoring 1,000 hospital patients per week; infection rate 0.002 (0.2%). λ = np = 2 infections/week. Binomial with n=1000, p=0.002 → Poisson (λ=2) is an excellent approximation.

Poisson distribution applications in microbiology: CFU counting: if plating a 10⁻⁶ dilution and expecting 50 colonies per plate, the actual count follows a Poisson distribution with λ = 50. The coefficient of variation = SD/mean = √λ/λ = 1/√λ = 1/√50 ≈ 14% — explains why plate counts are inherently variable. MPNB (Most Probable Number): dilution series into presence/absence wells; Poisson probability of at least one organism in each tube used to estimate the original cell density. Multiplicity of infection (MOI): when infecting cells with viruses, if MOI = 3 (average 3 viruses per cell): Poisson predicts P(0 viruses in cell) = e⁻³ = 5% of cells uninfected. Plaque assay: plaques counted follow a Poisson distribution — explains optimal plate count range (20–200) for reliable estimation.