Definition: The exponential distribution models the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.

• Mean (Expected Value): $E(X) = \frac{1}{\lambda}$
• Variance: $\text{Var}(X) = \frac{1}{\lambda^2}$
• Memoryless Property: $P(X > s + t | X > s) = P(X > t)$. This means that if an event has not occurred for some time s, the probability of waiting an additional time t is the same as the probability of waiting time t from the start. This unique property means the distribution has no "memory" of past waiting time. For example, if a light bulb has been working for 10 hours, the probability it will work for one more hour is exactly the same as the probability a brand new bulb will work for one hour. Let's prove this mathematically: $$P(X > s + t | X > s) = \frac{P(X > s + t)}{P(X > s)} = \frac{e^{-\lambda(s+t)}}{e^{-\lambda s}} = e^{-\lambda t} = P(X > t)$$ This shows that waiting an additional time t after already waiting for time s has the same probability as waiting time t from the start.
• Support: $[0, \infty)$
• Right-skewed distribution
• Decreasing failure rate