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Exponential Distribution Calculator
Computes the density, the cumulative probability P(X ≤ x), the survival probability P(X > x) and optionally P(x < X ≤ x2) for an exponential distribution given its rate λ or its mean 1/λ, plus mean, median, variance and standard deviation.
When to use
You model the time between independent random events at a constant rate (time to failure, time between arrivals, radioactive decay) and need the probability of waiting at most or more than a given time.
Do not use when: You need the probability of a number of events in an interval (use poisson-distribution), the trials are discrete (use geometric-distribution), or the failure rate changes over time (Weibull).
Formula
f(x) = λ e^(−λx); P(X ≤ x) = 1 − e^(−λx); P(X > x) = e^(−λx); mean = 1/λ; median = ln(2)/λ; variance = 1/λ²
The exponential distribution is memoryless: P(X > s + t | X > s) = P(X > t). It is the continuous counterpart of the geometric distribution and the inter-arrival time of a Poisson process with rate λ.
Inputs
| Parameter | Type | Unit | Required | Description |
|---|---|---|---|---|
rate | number | no | Events per unit time. Give rate or mean (mean = 1/rate). Range: > 0 | |
mean | number | no | Average waiting time, alternative to rate. Range: > 0 | |
x | number | yes | Time (or distance) at which to evaluate the distribution, in the same units as 1/rate. Range: ≥ 0 | |
x2 | number | no | Optional upper value; when given, P(x < X ≤ x2) is also returned. Range: ≥ 0 |
Outputs
| Output | Type | Unit | Description |
|---|---|---|---|
rate | number | Rate used. | |
mean | number | 1/λ. | |
probability_density | number | λ·e^(−λx) (not a probability). | |
probability_at_most | number | 1 − e^(−λx): probability the event has occurred by x. | |
probability_more_than | number | e^(−λx): probability of waiting longer than x (survival function). | |
probability_between | number | e^(−λx) − e^(−λx2) (only when x2 is given). | |
median | number | ln 2 / λ: half of the waiting times are shorter than this. | |
variance | number | 1/λ². | |
std_dev | number | 1/λ (equal to the mean). |
Example
λ = 0.5 per hour, x = 2 hours: {"rate":0.5,"x":2} → {"rate":0.5,"mean":2,"probability_density":0.18394,"probability_at_most":0.632121,"probability_more_than":0.367879,"median":1.386294,"variance":4,"std_dev":2}
Mean 10 years, x = 5 years, x2 = 20: {"mean":10,"x":5,"x2":20} → {"rate":0.1,"probability_density":0.060653,"probability_at_most":0.393469,"probability_more_than":0.606531,"probability_between":0.471196,"median":6.931472,"variance":100}
GET https://tttkmbb.com/api/v1/calculate/exponential-distribution?rate=0.5&x=2
Machine access
- API:
GET https://tttkmbb.com/api/v1/calculate/exponential-distribution(query parameters) orPOSTwith a JSON body{"inputs": {...}} - Schema: https://tttkmbb.com/api/v1/calculators/exponential-distribution · Markdown: https://tttkmbb.com/statistics/exponential-distribution.md · JSON definition: https://tttkmbb.com/statistics/exponential-distribution.json
- MCP: server
https://tttkmbb.com/mcp, toolrun_calculator with calculator_id="exponential-distribution" - OpenAPI operationId:
calculate_exponential_distribution_probability - Freshness:
static. Authentication: none. Rate limit: fair use (see rate limits).
Sources
FAQ
Rate or mean?
Either: a mean time between failures of 10 years is a rate of 0.1 per year. If both are given they must agree (rate × mean = 1).
Why is the median less than the mean?
The distribution is right-skewed: many short waits and a few very long ones. The median is ln 2 ≈ 0.693 times the mean.
Related calculators
- Poisson Distribution Calculator — Number of events in an interval at the same rate.
- Half-Life Calculator — Exponential decay expressed as half-life.
- Geometric Distribution Calculator — Discrete-trial analogue.