Enter the rate parameter
CDF at
PDF at
Expected Value:
Variance:
Random Sample:
The exponential distribution is widely used to model the time between events. It is defined by the parameter λ (>0).
The PDF of an exponential random variable X (with parameter λ) is given by:
fX(x) = λe-λx, for x > 0 fX(x) = 0, otherwise
The CDF of X is:
FX(x) = (1 - e-λx), for x ≥ 0 FX(x) = 0, otherwise
The expected value (mean) and variance of X are:
E[X] = 1/λ
Var(X) = 1/λ²
The exponential distribution can be seen as the continuous version of the geometric distribution. It represents waiting times, like waiting for customers to arrive at a store.
A key property is the “memoryless” feature:
P(X > x + a | X > a) = P(X > x), for a, x ≥ 0
This means the probability of waiting an additional time x doesn’t depend on how long you have already waited.