# Modelling Occurence of Events, with some Exposure

This afternoon, an interesting point was raised, and I wanted to get back on it (since I did publish a post on that same topic a long time ago). How can we adapt a logistic regression when all the observations do not have the same exposure. Here the model is the following: ,

• the occurence of an event $Y_i^\star$ on the period $[0,1]$ is unobserved
• the occurence of an event $Y_i$ on $[0,E_i]$ is observed (as well as $E_i$)

If we assume that the ‘occurence of an event’ is the first occurence of a Poisson processus, we can prove that

$\mathbb{P}(Y^\star=0)=\mathbb{P}(Y^\star_{[0,1]}=0)=\mathbb{P}(Y^\star_{[0,E]}=0)\cdot \mathbb{P}(Y^\star_{[E,1]}=0)$

i.e. no event occur on $[0,1]$ if no event occur on $[0,E]$ and no event occur on $[E,1]$. Assuming independence between the two, we can prove that we have

$\mathbb{P}(Y^\star=0) = \mathbb{P}(Y=0)^E$

With words, it means that the probability of not having a claim in the first six months of the year is the square root of not have a claim over a year. Which makes sense.