Suppose I want to test whether or not code patches created on weekends have a greater bug rate than those created on weekdays. (We might guess that this is so because people who are at work on weekends are more hurried etc.)
If we follow the standard and model bug creation as a Poisson process, this means (I think) that we want to tell the probability that both data sets came from the same underlying distribution, i.e. $\lambda_1 = \lambda_2$.
How can I do this? One way I thought of is use MLE to find the parameter which best fits one distribution, and then test the likelihood that parameter generates the second distribution. Alternatively, I could do two regressions and then use a likelihood ratio test. The problem with both methods is they are not testing whether one single $\lambda$ underlies both sets, but rather if the best-fit for one is the best-fit for the other.