3. You watch a Poisson Process with rate 1 per unit time for a time period of length t.

(a) Plot, as a function of t, the chance that there are (strictly) more points in the second half of the time period than in the first half. You will need to experiment with the range over which you plot.

(b) Use the plot to guess the limits of this probability as t 0 and as t .

(c) Prove that your guess at 0 is correct. A careful proof for the limit at is hard but I will provide an outline of a proof in the solutions.