{
  "id": "poisson-distribution",
  "version": "50fe64ff6e82",
  "status": "published",
  "name": "Poisson Distribution Calculator",
  "question": "Poisson distribution: how likely is k?",
  "summary": "Computes Poisson probabilities P(X = k), P(X < k), P(X ≤ k), P(X > k), and P(X ≥ k) for a mean rate λ, with the mean, variance, and standard deviation.",
  "category": "statistics",
  "subcategory": "distributions",
  "url": "https://www.acalculator.org/statistics/poisson-distribution-calculator",
  "markdown": "https://www.acalculator.org/statistics/poisson-distribution-calculator.md",
  "kind": "function",
  "method": "P(X = k) = e^−λ × λᵏ ÷ k!; P(X ≤ k) = Σ P(X = i) for i = 0 … k; mean = variance = λ.",
  "assumptions": [
    "Events happen one at a time, independently, at a constant average rate λ over the interval.",
    "P(X ≤ k) and P(X ≥ k) come from the regularized incomplete gamma function, which equals the sum of the terms; each tail is computed directly, not as 1 minus the other, so tiny tails keep their precision."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "lambda": {
        "title": "Average rate (λ)",
        "description": "The average number of events in the interval, more than 0 and up to 10,000.",
        "type": "number",
        "exclusiveMinimum": 0,
        "maximum": 10000
      },
      "k": {
        "title": "Number of events (k)",
        "description": "The number of events to find the probability of, a whole number from 0 to 100,000.",
        "type": "integer",
        "minimum": 0,
        "maximum": 100000
      }
    }
  },
  "outputs": {
    "exactly": {
      "label": "P(X = k)",
      "description": "The probability of exactly k events.",
      "format": "number"
    },
    "below": {
      "label": "P(X < k)",
      "description": "The probability of fewer than k events.",
      "format": "number"
    },
    "atMost": {
      "label": "P(X ≤ k)",
      "description": "The probability of k or fewer events.",
      "format": "number"
    },
    "above": {
      "label": "P(X > k)",
      "description": "The probability of more than k events.",
      "format": "number"
    },
    "atLeast": {
      "label": "P(X ≥ k)",
      "description": "The probability of k or more events.",
      "format": "number"
    },
    "mean": {
      "label": "Mean (λ)",
      "description": "The expected number of events: λ.",
      "format": "number"
    },
    "variance": {
      "label": "Variance (λ)",
      "description": "The variance of the number of events: λ.",
      "format": "number"
    },
    "sd": {
      "label": "Standard deviation (√λ)",
      "description": "The square root of λ.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "lambda": 3,
      "k": 5
    },
    "outputs": {
      "exactly": 0.10081881344492453,
      "below": 0.815263244523772,
      "atMost": 0.9160820579686968,
      "above": 0.08391794203130322,
      "atLeast": 0.18473675547622806,
      "mean": 3,
      "variance": 3,
      "sd": 1.7320508075688772
    },
    "text": "With an average of 3, the chance of exactly 5 events is 0.100819, and of at most 5 is 0.916082."
  },
  "examples": [
    {
      "given": {
        "lambda": 3,
        "k": 5
      },
      "expect": {
        "exactly": 0.10081881344492448,
        "atMost": 0.9160820579686966,
        "below": 0.8152632445237721,
        "above": 0.08391794203130346,
        "atLeast": 0.18473675547622792,
        "sd": 1.7320508075688772
      },
      "source": "hand calculation in content.mdx: e^−3 × 3⁵ ÷ 5! = 243 e^−3 ÷ 120; NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.19 Poisson Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm"
    },
    {
      "given": {
        "lambda": 2,
        "k": 0
      },
      "expect": {
        "exactly": 0.1353352832366127,
        "below": 0,
        "atLeast": 1,
        "atMost": 0.1353352832366127
      },
      "source": "hand calculation in content.mdx: e^−2 = 0.135335; NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.19 Poisson Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm"
    },
    {
      "given": {
        "lambda": 1,
        "k": 2
      },
      "expect": {
        "exactly": 0.18393972058572117,
        "atMost": 0.9196986029286058,
        "above": 0.08030139707139419
      },
      "source": "hand calculation in content.mdx: e^−1 (1 + 1 + 1/2) = 2.5 e^−1; NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.19 Poisson Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm"
    },
    {
      "given": {
        "lambda": 0.5,
        "k": 10
      },
      "expect": {
        "exactly": 1.6322616219566209e-10,
        "atLeast": 1.7096700293489033e-10
      },
      "source": "hand calculation in content.mdx: e^−0.5 × 0.5¹⁰ ÷ 10!, summed over k ≥ 10 in Python; NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.19 Poisson Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm"
    },
    {
      "given": {
        "lambda": 100,
        "k": 100
      },
      "expect": {
        "exactly": 0.039860996809147134,
        "atMost": 0.5265621985299984
      },
      "source": "hand calculation in content.mdx: e^−100 × 100¹⁰⁰ ÷ 100!, in logs; NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.19 Poisson Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm"
    }
  ],
  "sources": [
    "NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.19 Poisson Distribution (probability mass function, cumulative distribution function, mean λ, standard deviation √λ), retrieved 2026-10-01. https://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm"
  ],
  "related": [
    "binomial-distribution",
    "normal-distribution",
    "probability",
    "expected-value"
  ],
  "changelog": []
}