{
    "name": "Exponential Distribution Calculator",
    "url": "https://calculators.geekmidst.com/calculators/exponential-distribution-probability-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/exponential-distribution-probability-calculator.md",
    "category": "Statistics & probability",
    "description": "Find the probability that a waiting time or lifetime is below, above or between given values for an exponential distribution, from the rate or the mean.",
    "summary": "The exponential distribution describes the waiting time between events that happen at a constant average rate, such as the time to the next customer or to a random failure.",
    "also_called": [
        "Exponential CDF calculator",
        "Waiting time calculator"
    ],
    "applies_to": "Worldwide",
    "last_updated": "2026-10-29",
    "computation": "Results are computed in the visitor’s browser on the HTML page. This API describes the calculator; it does not compute results.",
    "inputs": [
        {
            "name": "param",
            "label": "I know the",
            "type": "segmented",
            "default": "rate",
            "options": {
                "rate": "Rate (λ)",
                "mean": "Mean time"
            }
        },
        {
            "name": "rate",
            "label": "Rate (λ), events per unit of time",
            "type": "number",
            "default": "0.5",
            "shown_when": {
                "param": "rate"
            }
        },
        {
            "name": "mean_time",
            "label": "Mean time between events (1/λ)",
            "type": "number",
            "default": "2",
            "shown_when": {
                "param": "mean"
            }
        },
        {
            "name": "mode",
            "label": "Probability that the time is",
            "type": "select",
            "default": "below",
            "options": {
                "below": "At most x: P(X ≤ x)",
                "above": "More than x: P(X > x)",
                "between": "Between a and b"
            }
        },
        {
            "name": "x",
            "label": "Time (x)",
            "type": "number",
            "default": "2",
            "shown_when": {
                "mode": [
                    "below",
                    "above"
                ]
            }
        },
        {
            "name": "a",
            "label": "From (a)",
            "type": "number",
            "default": "1",
            "shown_when": {
                "mode": "between"
            }
        },
        {
            "name": "b",
            "label": "To (b)",
            "type": "number",
            "default": "3",
            "shown_when": {
                "mode": "between"
            }
        }
    ],
    "outputs": [
        {
            "key": "probability",
            "label": "Probability",
            "format": "number"
        },
        {
            "key": "percent",
            "label": "As a percentage",
            "format": "percent"
        },
        {
            "key": "p_text",
            "label": "Probability, full precision",
            "format": "text"
        },
        {
            "key": "mean",
            "label": "Mean (1/λ)",
            "format": "number"
        },
        {
            "key": "median",
            "label": "Median (ln 2/λ)",
            "format": "number"
        },
        {
            "key": "rate",
            "label": "Rate λ",
            "format": "number"
        }
    ],
    "formula": {
        "text": "P(X ≤ x) = 1 − e^(−λx);  P(X > x) = e^(−λx);  P(a < X ≤ b) = e^(−λa) − e^(−λb);  mean = 1/λ, median = ln 2/λ, SD = 1/λ",
        "variables": {
            "λ": "rate of events per unit time",
            "x, a, b": "times in the same unit as the rate"
        }
    },
    "worked_example": {
        "inputs": {
            "param": "rate",
            "rate": "0.5",
            "mean_time": "2",
            "mode": "below",
            "x": "2",
            "a": "1",
            "b": "3"
        },
        "steps": [
            "λ = 0.5 per minute, so the mean wait is 1 ÷ 0.5 = 2 minutes.",
            "P(X ≤ 2) = 1 − e^(−0.5 × 2) = 1 − e⁻¹ = 0.632121.",
            "Median = ln 2 ÷ 0.5 = 1.386 minutes."
        ],
        "result": "With customers arriving at 0.5 per minute, there is a 63.2% chance the next one arrives within 2 minutes."
    },
    "interpretation": "The exponential distribution is memoryless: the chance of waiting another x minutes does not depend on how long you have already waited. The median is shorter than the mean, because a few long waits pull the mean up.",
    "assumptions": [
        "Events happen independently at a constant rate (a Poisson process), and time is measured in the same unit as the rate."
    ],
    "limitations": [
        "Lifetimes of parts that wear out (rising failure rate) or that have early failures are better described by a Weibull distribution."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "How is this linked to the Poisson distribution?",
            "answer": "If counts per interval are Poisson with rate λ, the gaps between events are exponential with the same λ."
        },
        {
            "question": "Why is the probability of waiting longer than the mean only 36.8%?",
            "answer": "P(X > 1/λ) = e⁻¹ = 0.368 for any rate, because the distribution is skewed towards short waits."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Poisson Probability Calculator",
            "url": "https://calculators.geekmidst.com/calculators/poisson-probability-calculator"
        },
        {
            "name": "Normal Distribution CDF Calculator",
            "url": "https://calculators.geekmidst.com/calculators/normal-distribution-cdf-calculator"
        },
        {
            "name": "Probability Calculator",
            "url": "https://calculators.geekmidst.com/calculators/probability-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}