Uniform Distribution: Continuous and Discrete for Exam P
Review both continuous and discrete uniform distributions and their Exam P applications.
Continuous Uniform Distribution
If X ~ Uniform(a, b), the PDF is f(x) = 1/(b - a) for a <= x <= b. The CDF is F(x) = (x - a)/(b - a). The mean is (a + b)/2 and the variance is (b - a)^2 / 12. The MGF is M(t) = (e^(tb) - e^(ta)) / (t(b - a)).
The uniform distribution models complete uncertainty within an interval. In insurance, it might model the time of an accident within a year (Uniform(0, 1)) or the location of damage along a structure of known length.
Discrete Uniform Distribution
If X takes values 1, 2, ..., n with equal probability 1/n, then X follows a discrete uniform distribution. The mean is (n + 1)/2 and the variance is (n^2 - 1)/12. This distribution arises in Exam P when all outcomes are equally likely, such as rolling a fair die or randomly selecting a policy number.
For integer-valued uniform on {a, a+1, ..., b}, the mean is (a + b)/2 and the variance is ((b - a + 1)^2 - 1)/12.
Exam P Applications
Uniform distributions often appear in "mixed" problems. For example, given that a loss is uniformly distributed on (0, theta) and theta itself has some distribution, you may be asked to find the unconditional distribution or expected value of the loss. The double expectation formula E[X] = E[E[X|Theta]] = E[Theta/2] is typically the fastest approach. Problems involving order statistics from a uniform sample are also common on Exam P.