CAS Exam 1Probability (P)

The number of claims from a policy in a year follows a Poisson distribution with mean 2. Calculate the probability that the policy produces at least two claims in a year.

A loss $X$ is exponentially distributed with mean 1,000. A policy pays the loss in excess of a 500 deductible. Calculate the expected payment per loss.

A random variable has moment generating function $M(t)=(1-2t)^{-3}$ for $t<1/2$. Calculate its variance.

Claim counts follow a negative binomial distribution with parameters $r=4$ and $\beta=1.5$ (mean $=r\beta$, variance $=r\beta(1+\beta)$). Calculate the variance of claim counts.

Aggregate annual claims for a book are approximately normal with mean 500 and standard deviation 100. Calculate the probability that aggregate claims exceed 650.