exponential probability distribution calculator

Use this calculator to evaluate probabilities, density, intervals, and percentiles for an exponential random variable with rate parameter λ.

The exponential distribution models waiting time between events in a Poisson process (for example: time between incoming support tickets, customer arrivals, or hardware failures).

Calculation Type

Rate parameter λ (> 0)

x (time/value, x ≥ 0)

What is the Exponential Distribution?

The exponential distribution is a continuous probability distribution often used to model waiting times. If events happen randomly but at a constant average rate, then the time between consecutive events follows an exponential pattern.

Common use cases include queueing theory, reliability engineering, risk analysis, network traffic modeling, and service operations. It is tightly connected to the Poisson process.

Core Formulas

PDF (density): f(x) = λe^-λx, for x ≥ 0
CDF: P(X ≤ x) = 1 - e^-λx
Survival function: P(X > x) = e^-λx
Interval probability: P(a < X ≤ b) = e^-λa - e^-λb
Percentile (inverse CDF): x = -ln(1-p)/λ
Mean: E[X] = 1/λ, Variance: Var(X)=1/λ²

How to Use This Calculator

Step-by-step

Choose a calculation type from the dropdown menu.
Enter the rate parameter λ.
Enter the required value(s): x, or interval bounds a and b, or probability p.
Click Calculate to see the result and the formula used.

Interpreting λ

The rate parameter λ is the average number of events per unit time. If λ is large, events occur more frequently, so waiting times are usually shorter. If λ is small, waiting times are usually longer.

Practical Example

Suppose a system receives critical alerts at an average rate of 2 per hour. Then λ = 2. What is the chance the next alert arrives within 30 minutes (0.5 hours)?

Use the CDF formula: P(X ≤ 0.5) = 1 - e^-2(0.5) = 1 - e^-1 ≈ 0.6321. So there is about a 63.21% chance the next alert arrives within half an hour.

Memoryless Property (Why Exponential is Special)

The exponential distribution is memoryless: P(X > s+t | X > s) = P(X > t). In plain language: if you've already waited for time s, the additional waiting time behaves as if you were starting over.

This property makes exponential models especially useful in Markov chains, stochastic processes, and simplified reliability models.

When to Use (and Not Use) It

Use exponential when:

Events occur independently.
The event rate is approximately constant over time.
You are modeling time between events.

Consider alternatives when:

The hazard rate changes with time (try Weibull or log-normal).
Data show strong seasonality or trend.
You model counts directly (Poisson may be more direct).

FAQ

Is the output a probability or density?

It depends on the selected calculation. CDF, survival, and interval outputs are probabilities. PDF output is a density value, not a direct probability.

Can x be negative?

No. For exponential distributions, support starts at 0. Values below 0 are invalid.

What unit should I use?

Any unit is fine (minutes, days, cycles), as long as you use it consistently for both λ and input values.