Are waiting times always going to be exponentially distributed?

Question

I'm studying for CAS/SOA Exam P/1 and a question I have here is:

We have a portfolio of $20$ insurance policies. The number of claims per policy in a $3$-month period has a Poisson distribution with mean $\frac12$. It is assumed that all of the policies in the portfolio are independent. What is the probability that there is a wait of more than $\frac12$ month before a claim is made by any policy in the whole portfolio?

Now the solution says that the mean for one month is $\frac{10}{3}$ which I understand where that comes from but then it says that the waiting time is exponentially distributed. Is that usually how waiting time problems are? Exponentially distributed if it's not said?

Answer 1

Not all waiting times are exponential, but the exponential is the waiting time distribution for the Poisson process. http://en.wikipedia.org/wiki/Exponential_distribution

Answer 2

Only exponential distributions are memoryless, and that means the probability of a claim in the next minute, given the history of claims to date, does not depend on the history. Are you more likely, or less likely, to get a claim in the next minute, given that there was one from some other geographic locale 10 minutes ago, then you would be if there hadn't been any claims in the past week? If the timing of those other claims has no bearing, then the distribution must be exponential.