$f(x) = \frac{1}{\theta}\,e^{-x/\theta}, \quad x > 0$ $\text{where } \theta > 0 = \text{mean (scale parameter)}$ The Exponential Distribution $X \sim \text{Exp}(\theta)$ is a continuous distribution on $(0, \infty)$ commonly used to model the time between events or the size of insurance losses. Its mean is $E[X] = \theta$, variance is $\text{Var}(X) = \theta^2$, and CDF is $F(x) = 1 - e^{-x/\theta}$. Its defining property is **memorylessness**: $P(X > s + t \mid X > s) = P(X > t)$, making it the unique continuous distribution with no aging effect. > [!example]- Probability a Loss Exceeds the Deductible {💡 Example} > Ground-up losses follow $X \sim \text{Exp}(\theta = 500)$. A deductible of $d = 300$ applies. What proportion of losses result in a claim payment? > > > [!answer]- Answer > > A payment is triggered only when $X > 300$: > > $P(X > 300) = 1 - F(300) = e^{-300/500} = e^{-0.6} \approx 0.5488$ > > About 54.9% of losses exceed the deductible and generate a payment.