I am trying to predict the number of events happening with the next time window, given the current values of some input variables. I am trying to pick up a good family of distributions to describe this data. Given the nature of the problem, Poisson distribution seems like a good idea.
When I plot histograms of data for different values of input variables, I see that there's a lot of mass at 0 (around 90-95% if I don't condition on anything), and the distribution for non-zero values looks like an exponential: the probability mass function is gradually decreasing. That already makes the Poisson distribution assumption questionable.
Furthermore, for the Poisson distribution the mean equals variance. In my case, however, if I plot the mean and variance for different values of conditional variables, I see a linear relation between them: variance = const * mean, where const is very high. Hence, again, Poisson distribution does not seem to be a good choice to fit this data. Which family of distributions would you suggest?