Recall that the poisson distribution is a discrete probability distribution that can model the number of times an event occurs in a given time or space interval. Suppose that an event can occur 1, 2, ... y times in an interval. The average number of events that occur in that interval are denoted 1 (not to be confused with eigen values), which is called the event rate or rate parameter. Thus, for a poisson distribution, the probability of observing y events in said interval is given by P(y; ) = ei a y!
With that in mind, consider the following problem. Given bicycle bridge crossing data (i. e. counts of the number of bicycles that cross a given bridge), the task is to build a model that estimates the number of bicycle crossings in a day given the High Temp (°F), the Low Temp (°F) and the Precipitation for that day.
a) Is the poisson distribution in the exponential family? If so, construct a GLM to solve this problem.
b) What are the natural parameter of the distribution, the sufficient statistic, the log partition function, and b(y).
c) What is the canonical response function for this problem?