Exponential Distribution

Probability · Axiom Academy

LESSON Exponential Distribution Modeling the time between events in a Poisson process 1. Probability Density Function The exponential PDF starts at λ when x = 0 and decays exponentially. The rate parameter λ determines how quickly the function decreases. Higher λ means events occur more frequently, so shorter waiting times are more likely. The function is only defined for x ≥ 0 since we cannot have negative waiting times. 2. Cumulative Distribution Function The CDF shows the probability that the waiting time is at most x. It starts at 0 and asymptotically approaches 1 as x increases, representing the increasing likelihood that the event has occurred by time x. 3. Expected Value and Variance The expected waiting time is the reciprocal of the rate parameter: 1/λ. This makes intuitive sense - if events occur at rate λ per unit time, the average waiting time between events is 1/λ. The variance also equals 1/λ², showing that the spread of waiting times increases as the rate decreases.

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