Lecture 5 : The Poisson Distribution Jonathan

In such situations we are often interested in whether the events occur randomly in time or space. Consider the Babyboom dataset we saw in Lecture 2. The birth times of the babies throughout the day are shown in Figure 1. If we divide up the day into 24 hour intervals and count the number of births in each hour we can plot the counts as a histogram in Figure 2. How does this compare to the histogram of counts for a process that isn’t random? Suppose the 44 birth times were distributed in time as shown in Figure 3. The histogram of these birth times per hour is shown in Figure 4. We see that the non-random clustering of events in time causes there to be more hours with zero births and more hours with large numbers of births than the real birth times histogram. This example illustrates that the distribution of counts is useful in uncovering whether the events might occur randomly or non-randomly in time (or space). Simply looking at the histogram isn’t sufficient if we want to ask the question whether the events occur randomly or not. To answer this question we need a probability model for the distribution of counts of random events that dictates the type of distributions we should expect to see.