# Distribution of the second largest event

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.

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If an event is Poisson distributed, does that mean it can occur more than once? – Henry Oct 23 '11 at 8:13
Yes, all events may occur more than once – fghftd Oct 23 '11 at 9:17
I presume that by the second largest event you mean the second most severe event. Since the events all occur arbitrarily often over time, with varying severities, it doesn't make sense to speak of "the" second most severe event without saying something about the time span over which the events are compared. By the distribution of the second largest event, do you mean its severity distribution? I'd expect that you'd have to say something about $F_k(T)$ to say anything useful about that. – joriki Oct 23 '11 at 9:23
Yes by second largest event i mean the second most severe outcome of any event. I actually know the distribution so it is given by D(T)=e^{\lambda(1-F(T))}(1+\lambda(1-F(T)) where F(T)=P(event\: severity<T|exactly\: one\: event\: occurs) butI do not have the derivation for this – fghftd Oct 23 '11 at 9:41

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 $e^{-\lambda(1-F(t))} (1 + \lambda (1 - F(t)))$.