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.

The joint density of $X$ and $Y$ is $f(x,y)=2$ for $0<x<y<1$ and zero otherwise. Calculate $E[X]$.

30% of drivers are high-risk with an annual accident probability of 0.20; the remaining 70% are low-risk with probability 0.05. A randomly chosen driver has an accident this year. Calculate the probability the driver is high-risk.

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

Let $X_1,\dots,X_n$ be i.i.d. Uniform$(0,\theta)$. Derive the distribution of $M=\max_i X_i$ and its expected value. Comment on whether $M$ is an unbiased estimator of $\theta$ and propose a correction.

Auto policyholders are classified as Preferred (50% of book, annual claim probability 0.02), Standard (30%, probability 0.05), or Substandard (20%, probability 0.10). A policyholder has a claim this year. Calculate the probability the policyholder is Substandard.

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.

The joint density of $X$ and $Y$ is $f(x,y)=x+y$ for $0<x<1,\,0<y<1$. Calculate $\operatorname{Cov}(X,Y)$.

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

Let $X$ and $Y$ be independent Exponential($\theta$) random variables (same $\theta$). Let $R=X/(X+Y)$. Derive the distribution of $R$ and interpret the result in terms of how loss dollars split between two independent, identically distributed claimants sharing a fixed total.