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Continuous Probability Tool

Exponential Distribution Calculator

Calculate exponential distribution probabilities using a rate parameter or mean. Find probability density, cumulative probability, survival probability, mean, variance, standard deviation, median, and quantiles instantly.

PDF & CDF Waiting-Time Model Percentiles Instant Results

Enter Distribution Values

Set the rate and X value for the calculation.

Choose whether you know the rate or mean.
Rate must be greater than zero.
Time or nonnegative value to evaluate.
Calculate the X value for a chosen percentile.

Interval Probability

Calculate P(c ≤ X ≤ d).

The exponential distribution is defined for X ≥ 0 and λ > 0. It is commonly used to model waiting times between independent events that occur at a constant average rate.
Probability Guide

What Is the Exponential Distribution?

The exponential distribution is a continuous probability distribution commonly used to model the amount of time between events occurring at a constant average rate. It is defined by a positive rate parameter called lambda (λ).

Examples include mathematical models for waiting time between arrivals, component lifetimes under a constant hazard assumption, and intervals between events in a Poisson process.

What Does Lambda Mean?

Lambda represents the average event rate per unit of time or measurement. A larger λ produces shorter expected waiting times, while a smaller λ produces longer expected waiting times.

How It Works

How to Use the Exponential Distribution Calculator

01

Enter λ or Mean

Choose whether you know the rate λ or the mean waiting time. The calculator automatically converts between them.

02

Enter X or an Interval

Provide a nonnegative X value for PDF and CDF calculations or enter c and d to calculate an interval probability.

03

View Statistics

Get PDF, CDF, survival probability, mean, variance, standard deviation, median, and the requested percentile.

Example: If λ = 0.5 events per unit time, the mean waiting time is 1 / 0.5 = 2 units. The probability of waiting no more than 3 units is 1 − e−0.5×3.
Distribution Formulas

Exponential Distribution PDF and CDF

Probability Density Function

The exponential probability density function is: f(x) = λe−λx for x ≥ 0. For negative values of x, the probability density is zero.

Cumulative Distribution Function

The cumulative probability of observing a value less than or equal to x is: F(x) = 1 − e−λx .

Survival Function

The probability that the random variable is greater than or equal to x is: S(x) = e−λx .

Distribution Statistics

Mean, Variance, Median and Standard Deviation

The mean of an exponential distribution is: Mean = 1 / λ . The standard deviation is also equal to 1 / λ.

The variance is: Variance = 1 / λ² , while the median is: Median = ln(2) / λ .

Key Property

What Is the Memoryless Property?

One of the defining mathematical properties of the exponential distribution is memorylessness. Under this model, the conditional probability of waiting an additional amount of time does not depend on how long has already elapsed.

This makes the exponential distribution useful for processes with a constant hazard rate. However, it may not be appropriate for real-world processes where risk increases or decreases with age or elapsed time.

Comparison

Exponential vs Poisson Distribution

Feature Exponential Poisson
Data Type Continuous Discrete
Measures Time or distance between events Number of events
Parameter Rate λ Expected event count
Typical Question How long until the next event? How many events occur?
Common Questions

Exponential Distribution Calculator FAQs

It is commonly used to model waiting times between independent events that occur at a constant average rate, including certain queueing, reliability, and Poisson-process models.
Lambda is the rate parameter. It represents the expected number of events per unit of time or measurement in the underlying constant-rate model.
The mean is calculated as 1 divided by lambda. For example, if λ = 0.25, the mean is 4 units.
No. The exponential distribution has support from zero to positive infinity, so x must be zero or greater.
It is the survival probability, or the probability that the random variable is at least x. For the exponential distribution it equals e raised to the power −λx.
The median is ln(2) divided by lambda. Half of the distribution's probability lies below this value and half lies above it.
For the exponential distribution, the hazard rate is constant and equals lambda. This constant hazard is directly related to the distribution's memoryless property.
The calculations on this page run directly in your browser using JavaScript, so the values do not need to be sent to a calculation server.