SOA PProbability

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.

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.

Annual claim totals for a book of business are normally distributed with mean 1,000 and standard deviation 150. Calculate the probability that claims exceed 1,200.