Distribution of the second largest event

Question

Consider M events that are all independent and poisson distributed in occurence with individual frequencies $\{\lambda_{k}\}_{k=1}^{M}$. Once they occur, they occur with a certain severity, event $k$ has severity distribution $F_{k}(T)$. I would like to know the distribution of the second largest event.

Answer 1

I suspect what is meant is something like this. Each event that occurs has a type (1 to $M$) and a severity $S$, such that the probability of any given occurrence being of type $k$ is $\lambda_k/\lambda$, and the cumulative distribution function of severity for events of type $k$ is $F_k(t)$, these being independent of whatever other events occur. Thus the combined cumulative distribution function of severity for each occurrence is $F(t) = \sum_{k=1}^M \frac{\lambda_k}{\lambda} F_k(t)$. The number of events that occur is a Poisson random variable with parameter $\lambda$, so the number of events of severity $> t$ that occur is a Poisson random variable with parameter $\lambda (1 - F(t))$. The probability that at most one event of severity $> t$ occurs (i.e. that the severity of the second most severe occurrence, if any, is at most $t$), is $e^{-\lambda(1-F(t))} (1 + \lambda (1 - F(t)))$.