The waiting-time law that forgets how long you already waited
Exponential waiting times model gaps between events in a Poisson stream—calls, machine failures, fabric defects—at a constant average rate. Mean and standard deviation both equal one over the rate λ. The shocking property: remaining wait, given you have already waited, looks like a fresh start.
In probability and statistics the exponential distribution—also called the negative exponential—describes gaps between events in a Poisson point process—arrivals that keep coming, independently, at a steady average tempo. That gap might be minutes between machine faults or centimeters along a roll of cloth. It is a special case of the gamma distribution, the continuous analogue of the geometric distribution, and is famous for memorylessness. It is not the same thing as the exponential family of distributions, a broad class that also includes normals, binomials, gammas, and Poissons.
With rate parameter λ greater than zero, the density is λ e to the minus λx for x at least zero (and zero otherwise); the cdf is one minus e to the minus λx on that support. Writers often use the scale β equals one over λ, which equals the mean. For X ~ Exp(λ), the expectation is one over λ and the variance is one over λ squared, so the standard deviation matches the mean. Moments satisfy E[X^n] equals n factorial over λ to the n; the median is ln(2)/λ, which is strictly less than the mean. The interquartile range is ln(3)/λ.
Memorylessness says that the chance of waiting more than s plus t, given you have already waited more than s, equals the chance of waiting more than t from scratch. If nothing has happened after thirty seconds, the chance of needing at least ten more seconds equals the unconditional chance of waiting more than ten seconds from the start. Among probability laws, only the exponential (continuous) and geometric (discrete) are memoryless; the exponential is therefore the only continuous distribution with a constant failure rate. That is why it is the default model when “wear-out” should not matter and only a constant hazard does.
Source: Exponential distribution
