SOA PProbability

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.

$X$ and $Y$ are independent exponential random variables with $X\sim\text{Exp}(1)$ (mean 1) and $Y\sim\text{Exp}(2)$ (mean $1/2$, rate 2). Calculate $P(X<Y)$.

The number of trials until the first success is geometric with $p=0.25$ (support $1,2,3,\dots$). Calculate $P(N>5)$.

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.

A binomial random variable has $n=100$ trials with $p=0.02$. Calculate its variance.