# 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.

- Calculator id: `exponential-distribution` · Category: Statistics & Probability (`statistics`) · Tool name: `calculate_exponential_distribution_probability`
- Canonical page: https://tttkmbb.com/statistics/exponential-distribution · This document: https://tttkmbb.com/statistics/exponential-distribution.md · JSON definition: https://tttkmbb.com/statistics/exponential-distribution.json

## Purpose

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.

**Use when:** 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).

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `rate` | number |  | optional | Events per unit time. Give rate or mean (mean = 1/rate). (> 0) |
| `mean` | number |  | optional | Average waiting time, alternative to rate. (> 0) |
| `x` | number |  | required | Time (or distance) at which to evaluate the distribution, in the same units as 1/rate. (min 0) |
| `x2` | number |  | optional | Optional upper value; when given, P(x < X ≤ x2) is also returned. (min 0) |

## Output

| Field | 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). |

## 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.

## Data Sources

- NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.7 Exponential Distribution — https://www.itl.nist.gov/div898/handbook/eda/section3/eda3667.htm (government, retrieved 2026-09-24)
- Wikipedia – Exponential distribution — https://en.wikipedia.org/wiki/Exponential_distribution (reference, retrieved 2026-09-24)

Data freshness: `static`. Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output.

## API

- `GET https://tttkmbb.com/api/v1/calculate/exponential-distribution?x=…`
- `POST https://tttkmbb.com/api/v1/calculate/exponential-distribution` with JSON body `{"inputs": {…}}`
- Response: unified envelope (`success`, `request`, `result.values`, `result.units`, `sources`, `freshness`, `timestamp`, `next_actions`, `links`); see https://tttkmbb.com/docs/response-format.md
- Schema: https://tttkmbb.com/api/v1/calculators/exponential-distribution · OpenAPI operationId `calculate_exponential_distribution_probability` in https://tttkmbb.com/openapi.json
- Authentication: none. Rate limit: fair use, see https://tttkmbb.com/docs/rate-limits.md.

## MCP

- Server: `https://tttkmbb.com/mcp` (Streamable HTTP, JSON-RPC 2.0, no auth)
- Tool:  `run_calculator` with `{"calculator_id": "exponential-distribution", "inputs": {…}}`

## Example

- λ = 0.5 per hour, x = 2 hours: inputs `{"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: inputs `{"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
```

## Limitations

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). 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. All values are computed from the formula above; no measurement or live data is involved.

## 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

- [Poisson Distribution Calculator](https://tttkmbb.com/statistics/poisson-distribution.md) — Number of events in an interval at the same rate.
- [Half-Life Calculator](https://tttkmbb.com/chemistry/half-life.md) — Exponential decay expressed as half-life.
- [Geometric Distribution Calculator](https://tttkmbb.com/statistics/geometric-distribution.md) — Discrete-trial analogue.