Exponential Distribution Calculator

The exponential distribution models the time between events in a Poisson process — for example, the time until a radioactive particle decays or the time between customer arrivals. It is characterized by a single parameter λ (the rate), and its PDF is f(x) = λe^(−λx) for x ≥ 0.

λ (lambda) is the average number of events per unit of time. The mean of the distribution equals 1/λ, so a higher λ means events happen more frequently and the distribution is more concentrated near zero.

P(X < x) is given by the cumulative distribution function (CDF): F(x) = 1 − e^(−λx). This gives the probability that the random variable X takes a value less than x.

The PDF (probability density function) f(x) = λe^(−λx) gives the density of probability at a specific point x, while the CDF F(x) = 1 − e^(−λx) gives the cumulative probability that X is less than or equal to x. For continuous distributions, P(X = x) = 0 exactly.

Use the interval probability formula: P(x₁ < X < x₂) = F(x₂) − F(x₁) = e^(−λx₁) − e^(−λx₂). Enter both bounds in the Lower Bound and Upper Bound fields of this calculator to get the result automatically.

The mean is μ = 1/λ and the variance is σ² = 1/λ². The standard deviation equals 1/λ, which means the standard deviation is always equal to the mean — a unique property of the exponential distribution.

Yes. The exponential distribution is the only continuous distribution with the memoryless property: P(X > s + t | X > s) = P(X > t). This means the probability of waiting an additional time t is the same regardless of how long you have already waited.

If events occur according to a Poisson process with rate λ, the time between consecutive events follows an Exponential(λ) distribution. They are complementary: Poisson counts events in a fixed interval, while the exponential models the waiting time between events.