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.

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

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

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