# Exponential Distribution Calculator

Free exponential distribution calculator. Calculate CDF, survival function, and PDF for exponential distributions modeling time between events.

## What this calculates

Calculate probabilities for the exponential distribution, which models the time between independent events occurring at a constant rate. Find CDF, survival function, and PDF values.

## Inputs

- **Rate Parameter (λ)** — min 0.0001 — The rate of events per unit time (λ > 0). Also equals 1/mean.
- **X Value (time/distance)** — min 0 — The value to calculate the probability for.
- **Lower Bound (optional)** — min 0 — Lower bound for P(a < X < b).
- **Upper Bound (optional)** — min 0 — Upper bound for P(a < X < b).

## Outputs

- **P(X ≤ x)** — CDF: Probability that X is less than or equal to x.
- **P(X > x)** — Survival function: Probability that X exceeds x.
- **P(a < X < b)** — Probability that X falls between lower and upper bounds.
- **PDF f(x)** — The probability density at x.
- **Mean (1/λ)** — The expected value (average time between events).
- **Variance (1/λ²)** — The variance of the distribution.
- **Median** — The median: ln(2)/λ.

## Details

The exponential distribution models waiting times between events in a Poisson process.

PDF: f(x) = λe^(-λx) for x ≥ 0
CDF: F(x) = 1 - e^(-λx)
Survival: S(x) = e^(-λx) = P(X > x)

Key Properties

- Mean = 1/λ

- Variance = 1/λ²

- Median = ln(2)/λ

- Memoryless property: P(X > s+t | X > s) = P(X > t)

Applications: Time between customer arrivals, equipment failure times, radioactive decay.

## Frequently Asked Questions

**Q: What is the memoryless property?**

A: The exponential distribution is the only continuous distribution with the memoryless property: the probability of waiting at least t more units is the same regardless of how long you have already waited. Mathematically: P(X > s+t | X > s) = P(X > t). This means past waiting provides no information about future waiting.

**Q: How is the exponential distribution related to the Poisson distribution?**

A: If events follow a Poisson process with rate λ, the time between consecutive events follows an exponential distribution with the same rate λ. The Poisson counts events in a fixed interval; the exponential measures time between events.

**Q: How do I choose between rate (λ) and mean (1/λ) parameterization?**

A: Both are equivalent. If you know the average time between events (e.g., customers arrive every 5 minutes on average), the mean is 5 and λ = 1/5 = 0.2. If you know the rate (e.g., 3 events per hour), then λ = 3 and the mean is 1/3 hours.

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Source: https://vastcalc.com/calculators/statistics/exponential-distribution
Category: Statistics
Last updated: 2026-04-21