Exponential Distribution

A continuous random variable $X\ge 0$ follows an Exponential distribution with rate $\lambda >0$ if its density is $f(x)=\lambda e^{-\lambda x}$, with mean $1/\lambda$ and variance $1/{\lambda }^2$. It is the unique continuous distribution with the memoryless property, $P(X>s+t\mid X>s)=P(X>t)$ for all $s,t\ge 0$, and describes the waiting time between events in a Poisson process, making it the continuous counterpart to the discrete Geometric distribution.

If customer arrivals at a service desk follow a Poisson process at a rate of $\lambda =4$ per hour, the time between consecutive arrivals follows an Exponential($\lambda =4$) distribution, with a mean waiting time of $1/4$ hour (15 minutes) between arrivals -- and, by memorylessness, having already waited 10 minutes for the next customer doesn't shorten or lengthen the expected remaining wait.

• The Exponential distribution's memoryless property, $P(X>s+t\mid X>s)=P(X>t)$, is mathematically unique to it among continuous distributions, no aging, wear, or fatigue effect can be represented by an Exponential model, which is exactly why the Weibull distribution, which generalizes it, is preferred whenever failure rates change over time.
• The Exponential distribution is the continuous-time waiting analogue of the discrete Geometric distribution, both share the memoryless property, and the Exponential arises as the limiting case of a Geometric distribution as the time step shrinks to zero while the event rate is held fixed.
• A sum of $k$ independent Exponential($\lambda$) random variables follows an Erlang (Gamma with integer shape) distribution, making the Exponential distribution the fundamental building block of the broader Gamma family.