A Poisson process is one where events occur continuously and independently at a constant average rate (\(\lambda\)). The Exponential distribution describes the waiting time for the next event.
Examples: Time until the next customer arrives, time until a radioactive particle decays, time until a component fails (assuming constant failure rate), duration of telephone calls.
A continuous random variable \(X\) follows an Exponential distribution if it represents the waiting time for an event in a process where events occur at a constant average rate \(\lambda\).
Possible values for \(X\) are non-negative real numbers (\(X \ge 0\)).
Notation: \(X \sim \text{Exponential}(\lambda)\) or \(X \sim \text{Exp}(\lambda)\). (Sometimes parameterized by the mean \(\beta = 1/\lambda\), as \(\text{Exp}(\beta)\)). We'll use the rate parameter \(\lambda\).
The PDF shows that shorter waiting times are more likely, with the probability decreasing exponentially as the waiting time \(x\) increases:
$$
f(x | \lambda) = \begin{cases} \lambda e^{-\lambda x} & \text{if } x \ge 0 \ 0 & \text{if } x < 0 \end{cases}
$$
Where:
\(x\) is the waiting time (\(x \ge 0\)).
\(\lambda\) (lambda) is the rate parameter (average number of events per unit of time), \(\lambda > 0\).
\(e\) is Euler's number (\(e \approx 2.71828...\)).
The CDF\(F(x) = P(X \le x)\) gives the probability that the event occurs within time \(x\):
$$
F(x) = P(X \le x) = \begin{cases} 1 - e^{-\lambda x} & \text{if } x \ge 0 \ 0 & \text{if } x < 0 \end{cases}
$$
This is derived by integrating the PDF: \(\int_0^x \lambda e^{-\lambda t} dt\).
The Exponential distribution has a unique property among continuous distributions called memorylessness.
Mathematically: \(P(X > s + t | X > s) = P(X > t)\) for all \(s, t \ge 0\).
Intuition: If you've already waited \(s\) units of time for an event without it occurring, the probability that you'll have to wait an additional\(t\) units of time is exactly the same as the probability you would have had to wait \(t\) units from the very beginning. The process "doesn't remember" how long you've already waited.
Example: If component lifetime follows an Exponential distribution, knowing it has already survived for 100 hours doesn't change the probability distribution of its remaining lifetime. This is often unrealistic for mechanical components (which wear out) but can be applicable to certain electronic components or radioactive decay.
Poisson Distribution: Describes the number of events in a fixed interval, while Exponential describes the time between those events. They are intrinsically linked through the Poisson process.
[[10_Gamma_Distribution|Gamma Distribution]]: The Exponential distribution is a special case of the Gamma distribution where the shape parameter \(\alpha=1\) (\(Exp(\lambda) = Gamma(1, \lambda)\)). The Gamma distribution models the waiting time until the \(\alpha\)-th event.
[[08_Geometric_Distribution|Geometric Distribution]]: The discrete analogue of the Exponential distribution. It models the number of Bernoulli trials needed until the first success and also possesses a discrete version of the memorylessness property.
Used heavily in Reliability Engineering and Survival Analysis to model lifetimes or time-to-failure (though often more complex distributions like Weibull are needed if the failure rate isn't constant).
Used in Queueing Theory to model inter-arrival times and service times.