How to use

This calculator evaluates an exponential distribution with positive rate λ. It returns the density at x, CDF P(X≤x), survival probability P(X>x), interval probability, mean, variance, standard deviation and median. The model is commonly used for waiting times between independent events occurring at a constant average rate.

Enter a positive event rate λ.
Enter x and the interval endpoints a and b.
Calculate the density, cumulative and survival probabilities and summary properties.

How it is calculated

f(x)=λe^(−λx); F(x)=1−e^(−λx); E[X]=1/λ

Example

Example: with λ=0.5 and x=3, P(X>3)=e^-1.5≈0.22313, P(X≤3)≈0.77687 and the mean is 1/λ=2.

Important notes

The exponential model assumes a constant rate and memorylessness. If the hazard changes with time, another model may be more appropriate. Some references parameterize the distribution by scale β=1/λ instead of rate λ.

Frequently asked questions

What is the difference between PDF and CDF?

The PDF is a density; the CDF is the probability that X is at most x.

Is P(X=x) equal to the PDF?

No. For a continuous distribution, a single exact point has probability zero.

What is the mean?

Mean and standard deviation are both 1/λ, while variance is 1/λ².

Is this calculator free?

Yes. It is free to use and requires no account.

Exponential Distribution Calculator