How does the exponential distribution model waiting times between events?

The exponential distribution models the time elapsed between independent events that occur at a constant average rate. It is particularly useful for scenarios where smaller time intervals are more likely to occur than larger ones.1

Key characteristics include:

• Memorylessness: The probability of an event occurring in the future is independent of how much time has already passed. This assumes the average rate of events remains constant over time.1
• Mathematical Foundation: The probability density function is defined as $f(x)=\lambda e^{-\lambda x}$ for $x>0$, where $\lambda$ represents the average number of events per unit time.1
• Relationship to Other Models: While the Poisson distribution determines the number of events occurring in a fixed interval, the exponential distribution models the duration between those individual events.1

Because it assumes a constant rate, it is not suitable for processes where the likelihood of an event changes over time, such as mechanical systems that degrade with age.1