Exponential mean and variance calculator
At a rate of 2 events per hour, the average wait is 0.5 hours, the variance 0.25 and the standard deviation 0.5. Enter the rate λ of your exponential distribution and get all three moments at once. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Lambda
- 2
- Mean
- 0.5
- Variance
- 0.25
- Std
- 0.5
The exponential distribution is the classic continuous model for waiting times and inter-arrival gaps when events arrive at a constant average rate with no memory of the past. Once you fix the rate parameter λ, three moments fall out in closed form: the mean is one over λ, the variance is one over λ squared, and the standard deviation is again one over λ.
Rate parameterization and the three closed-form moments
Under the rate parameterization, a non-negative continuous random variable X follows Exp(λ) when its density is λ times e to the power of minus λx for x greater than or equal to zero, and zero otherwise. The rate λ is the expected number of events per unit time, so larger λ means shorter typical waits. Integrating x against that density yields the mean μ equals one over λ.
Integrating the squared deviation from the mean yields the variance σ squared equals one over λ squared. Taking the positive square root of the variance produces the standard deviation σ, which equals one over λ and therefore matches the mean in absolute scale—a distinctive property of the exponential family that students often use as a quick sanity check. Some textbooks prefer a scale parameterization with θ equals one over λ; in that writing the mean is θ, the variance is θ squared, and the standard deviation is θ.
The memoryless property says that the remaining wait given survival so far has the same distribution as a fresh wait, which is why the exponential appears in Poisson process inter-arrival times, radioactive decay waiting models, and simple reliability models with constant hazard. Because every output is a finite rational function of λ alone, the procedure is fully deterministic: the same rate always yields the same mean, variance, and standard deviation, with no Monte Carlo noise and no table look-up.
Where exponential moments show up in real work
Introductory probability and statistics courses constantly assign problems of the form: given an average of λ events per hour, what is the expected wait and how spread is it? Operations research and queueing sketches use exponential inter-arrival and service assumptions as the M/M building blocks; mean and variance of those waits feed capacity planning and simple Little-law arguments before heavier simulation.
Reliability and survival work often starts with a constant hazard rate λ, where mean time to failure is one over λ and the variance one over λ squared quantifies uncertainty around that MTTF under the pure exponential model. Product analytics and site reliability teams convert an observed events-per-minute rate into expected inter-event gap and spread when building dashboards or alert thresholds.
If you later need the exponential PDF or CDF at a point, maximum-likelihood estimation of λ from a sample, Poisson counts, geometric waiting in discrete trials, or a full descriptive summary of a numeric list, treat those as separate search intents and landings rather than overloading this exponential mean and variance calculator page.
What you can do with it
Homework and probability labs
Students and graders verify μ = 1/λ, σ² = 1/λ², and σ = 1/λ for textbook rates, including the hard rule that non-positive λ is invalid.
Queueing and reliability sketches
Engineers turn a service or failure rate into expected wait or MTTF plus variance before building heavier simulation models.
FAQ
What formulas does the calculator use?
For X ~ Exp(λ) under the rate parameterization: mean μ = 1/λ, variance σ² = 1/λ², and standard deviation σ = 1/λ. Scale-form textbooks should convert with λ = 1/θ first.
What happens if lambda is zero or negative?
The exponential distribution requires λ > 0; when the rate is not positive the moments are undefined. Validation failures are not charged.
Is this the rate or scale parameterization?
Rate only: pdf f(x) = λ e^{−λx}. Do not pass θ as lambda without converting.
What does it cost?
The browser tool is free and runs locally.
Why are the mean and the standard deviation the same?
Because for an exponential distribution both equal one divided by the rate. It is a distinctive property of this family, and a quick sanity check for homework: if your computed mean and standard deviation differ, recheck the arithmetic.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/stat/exponential-mean-variance \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"lambda":2}'const res = await fetch("https://api.kit.forhosting.com/stat/exponential-mean-variance", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"lambda": 2
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/stat/exponential-mean-variance",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"lambda": 2
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/stat/exponential-mean-variance", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"lambda":2}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"lambda":2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/stat/exponential-mean-variance", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"lambda": 2
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "stat.exponential_mean_variance",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
This calculator takes a single strictly positive finite rate, returns those three numbers with a short formula string, and rejects zero or negative λ with a clear invalid_input error so pipelines never invent silent nonsense.
Rate parameterization and the three closed-form moments
This capability uses rate form only: if your source states a scale θ, convert with λ equals one over θ before you call the API.
Input contract, validation, and response shape
A zero or negative rate fails with a message that the exponential distribution is undefined when λ is not greater than zero; those validation failures are not billed and do not invent a mean. On success the body echoes the cleaned lambda, returns mean as one over λ, variance as one over λ squared, std as one over λ, and a formula string restating μ equals one over λ, σ squared equals one over λ squared, and σ equals one over λ under the rate parameterization. Classic checks include λ equals one, which yields mean one, variance one, and std one; and λ equals two, which yields mean one half, variance one quarter, and std one half. Wire the free browser form for homework and quick sanity checks, then call the identical contract from scripts when you need bulk evaluation of service rates or failure rates. POST a JSON body with one required finite number named lambda: the rate parameter λ, which must be strictly greater than zero. The engine also accepts the aliases rate, lam, and the Unicode character λ, so scripts and classroom worksheets that already speak in those terms do not need a rename step; providing more than one of those keys at once fails validation so callers never silently pick the wrong field. Missing fields, empty strings, non-numeric tokens, NaN, and Infinity fail with invalid_input naming the field. Pricing for successful API calls is the published base of $0.002 per item.
Where exponential moments show up in real work
A single trusted oracle that returns cleaned JSON speeds grading, demo setup, and answer-key generation without re-deriving the integrals every time. Because this capability is a pure parse engine, API pricing is the published base of $0.002 per successful item and failed validation does not consume prepaid balance.
Integration use cases
Scripts and CI numeric fixtures
Pipelines and interview sandboxes call a deterministic exponential-moments contract instead of reimplementing 1/λ and 1/λ² in every language.
Technical questions
How do I compute exponential mean and variance with this calculator?
Send JSON with lambda (a finite positive rate) to stat.exponential_mean_variance. The response returns mean, variance, std, the cleaned lambda, and a formula string.