Exponential Distribution Calculator

Compute the probability density, cumulative probability, upper tail, mean, and variance of the exponential distribution from a rate or mean, with PDF and CDF charts.

Input

Enter the rate lambda (or mean 1/lambda) and a value x to compute the exponential distribution density, cumulative probability, upper tail, mean, and variance at once.

Value x

The point at which to evaluate. Enter a real number of 0 or more.

Rate lambda

The event rate per unit. Enter a real number greater than 0. The mean equals 1/lambda.

Result

Lower tail probability F(x) at x = 1, lambda = 0.5

0.39346934

Upper tail S(x)

0.60653066

Density f(x)

0.30326533

Mean 1/lambda

Variance 1/lambda squared

Probability density function f(x)

Cumulative distribution function F(x)

How it works

The exponential distribution is a continuous distribution with a rate parameter lambda (larger lambda means events occur more frequently) or mean 1/lambda, used to model waiting times between events in a Poisson process and component lifetimes.
The probability density function is f(x) = lambda e^(-lambda x) for x at least 0, decaying exponentially as x grows.
The cumulative distribution function (lower tail) is F(x) = 1 - e^(-lambda x), the probability that an event occurs by time x.
The upper tail (survival function) is S(x) = e^(-lambda x), the probability that no event occurs before x.
The mean is 1/lambda and the variance is 1/lambda squared. The exponential distribution is memoryless, meaning the remaining waiting time does not depend on how long you have already waited.

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