(R code and computing output for part b separately in machine-readable form.)

7. You have two independent Poisson processes. One generates red events and one generates blue events.

(a) Give a formula in terms of the two rate parameters for the probability that the number of red events is the same as the number of blue events in the first t time units? Also, give a formula for the probability of more red events than blue events in the same time period. I expect your answers to be infinite sums which you cannot do.

(b) Now suppose t = 1 and one of the rate parameters is equal to 1. Plot the answers to the previous part as a function of the other rate parameter.

(c) Submit a small Monte Carlo study to provide evidence that your answer to b) is correct.