$f(x) = \frac{1}{b - a}, \quad a < x < b$ The Continuous Uniform Distribution $X \sim \text{Unif}(a, b)$ assigns equal probability density to every point in the interval $(a, b)$, making it the continuous analogue of the discrete uniform. Its mean is $E[X] = (a+b)/2$, variance is $\text{Var}(X) = (b-a)^2/12$, and CDF is $F(x) = (x-a)/(b-a)$ for $a < x < b$. It is commonly used as a simple loss model when all outcomes in a range are equally plausible, and it has the memoryless property that makes conditional distributions on sub-intervals uniform. > [!example]- Expected Payment with Uniform Losses and a Deductible {💡 Example} > Ground-up losses $X \sim \text{Unif}(0, 1000)$. An ordinary deductible of $d = 300$ applies. Find $E[(X - 300)_+]$. > > > [!answer]- Answer > > Since losses are uniform, those above 300 are distributed $\text{Unif}(300, 1000)$. The probability of exceeding the deductible is $P(X > 300) = 700/1000 = 0.7$, and given $X > 300$ the expected excess is $(700)/2 = 350$. Therefore: > > $E[(X-300)_+] = P(X > 300) \times E[X - 300 \mid X > 300] = 0.7 \times 350 = 245$