Uniform Continuous Distribution

Study guides: Exam P-1, Exam MAS-I

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. It is commonly used as a simple loss model when all outcomes in a range are equally plausible.

$f(x) = \frac{1}{b - a}, \quad a < x < b$

$E[X] = (a+b)/2$ , $\text{Var}(X) = (b-a)^2/12$ , and $F(x) = (x-a)/(b-a)$ for $a < x < b$
Conditional distributions on sub-intervals are also uniform: given $X > d$ , $X$ is uniform on $(d, b)$
The uniform is not memoryless — the exponential is the only continuous density that is. For $\text{Unif}(0, 10)$ , $P(X > 8 \mid X > 5) = 2/5$ , but $P(X > 3) = 7/10$

Expected Payment with Uniform Losses and a Deductible Ground-up losses $X \sim \text{Unif}(0, 1000)$ . An ordinary deductible of $d = 300$ applies. Find $E[(X - 300)_+]$ .

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: $E[(X-300)_+] = P(X > 300) \times E[X - 300 \mid X > 300] = 0.7 \times 350 = 245$