What is the rate parameter lambda?

Lambda (λ) is the average number of events per unit of time or space. Its reciprocal (1/λ) is the mean inter-event time. Lambda must be strictly positive.

What does the PDF value represent?

The probability density function f(x) = λe^(−λx) gives the instantaneous likelihood density at x. It is not a probability itself but is used to compute probabilities over intervals.

What is the survival function?

The survival function S(x) = 1 − CDF = e^(−λx) gives the probability that the event has not yet occurred by time x. It is widely used in reliability and survival analysis.

Why is the exponential distribution called memoryless?

Given that an event has not occurred up to time t, the probability of it occurring in the next s units is the same as starting fresh. Formally: P(X > t+s | X > t) = P(X > s).

How is the median calculated?

The median is the value x where CDF = 0.5, which solves to x = ln(2) / λ ≈ 0.6931 / λ. It is always less than the mean (1/λ) because the distribution is right-skewed.

What is the variance of the exponential distribution?

The variance equals 1/λ². The standard deviation equals 1/λ, which is the same as the mean — a unique property of the exponential distribution.

Exponential Distribution Calculator

Name: Exponential Distribution Calculator
Availability: InStock
Author: Nham Vu

Enter the rate parameter lambda and a value x to get the PDF, CDF, survival function, mean, variance, and median of the exponential distribution.

Parameters

Rate Parameter λ (must be > 0)

Average number of events per unit time/space.

Value x (≥ 0)

The point at which to evaluate the distribution.

Quick Presets

Results at x

Enter parameters and press Calculate.

Distribution Statistics

Enter parameters and press Calculate.

Formulas

PDF:: f(x) = λ · e^−λx
CDF:: F(x) = 1 − e^−λx
Survival:: S(x) = e^−λx
Mean:: 1 / λ
Variance:: 1 / λ²
Median:: ln(2) / λ

Summary

Enter the rate parameter lambda and a value x to get the PDF, CDF, survival function, mean, variance, and median of the exponential distribution.

How it works

Enter a positive rate parameter lambda (λ) — the average number of events per unit time.
Enter a non-negative value x — the time or distance at which to evaluate the distribution.
The calculator instantly shows the PDF: λe^(−λx).
The CDF (probability that the event has occurred by time x) is shown as 1 − e^(−λx).
The survival function (probability the event has NOT yet occurred) equals e^(−λx).
Summary statistics — mean (1/λ), variance (1/λ²), and median (ln2/λ) — are displayed below.

Use cases

Modeling wait times in queuing systems such as call centers or service lines.
Reliability engineering: estimating the probability a component survives past time x.
Network analysis: computing packet inter-arrival probabilities.
Physics: radioactive decay probability calculations.
Finance: modeling time between rare market events.
Biology: analyzing time between mutations or cell divisions.
Statistics coursework: verifying manual calculations for homework or exams.
Quality control: predicting product failure rates.

Frequently Asked Questions

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Last updated: 2026-05-23 · Reviewed by Nham Vu