1 00:00:00,460 --> 00:00:03,970 In this lecture, we look in some depth into various 2 00:00:03,970 --> 00:00:06,690 properties of Poisson processes. 3 00:00:06,690 --> 00:00:10,030 These properties would be quite hard to study if one 4 00:00:10,030 --> 00:00:12,390 were to proceed just analytically by 5 00:00:12,390 --> 00:00:14,620 manipulating formulas. 6 00:00:14,620 --> 00:00:17,930 However, by using memorylessness and our 7 00:00:17,930 --> 00:00:20,650 intuitive understanding of what the Poisson process 8 00:00:20,650 --> 00:00:23,770 really is, they become quite simple. 9 00:00:23,770 --> 00:00:26,550 And the mathematical manipulations can be avoided 10 00:00:26,550 --> 00:00:29,690 almost entirely. 11 00:00:29,690 --> 00:00:33,150 We will start by arguing that the sum of independent Poisson 12 00:00:33,150 --> 00:00:36,080 random variables is Poisson. 13 00:00:36,080 --> 00:00:40,250 But we will then establish the much stronger statement that 14 00:00:40,250 --> 00:00:44,650 if we merge two independent Poisson processes, we, again, 15 00:00:44,650 --> 00:00:47,500 obtain a Poisson process. 16 00:00:47,500 --> 00:00:50,250 We will see that we can exploit this fact to solve 17 00:00:50,250 --> 00:00:54,130 some problems that would be quite difficult otherwise. 18 00:00:54,130 --> 00:00:58,820 Once more, intuitive reasoning is the key. 19 00:00:58,820 --> 00:01:02,040 Finally, we will spend some time discussing a phenomenon 20 00:01:02,040 --> 00:01:06,160 that goes under the name of random incidence. 21 00:01:06,160 --> 00:01:08,590 The Poisson process has been running. 22 00:01:08,590 --> 00:01:10,890 You show up at a certain time. 23 00:01:10,890 --> 00:01:14,110 You look at the size of the inter-arrival interval during 24 00:01:14,110 --> 00:01:15,940 which you show up. 25 00:01:15,940 --> 00:01:19,450 It turns out that this inter-arrival interval that 26 00:01:19,450 --> 00:01:22,810 you get to observe is not a typical one. 27 00:01:22,810 --> 00:01:25,260 It tends to be larger than the typical 28 00:01:25,260 --> 00:01:27,940 inter-arrival interval. 29 00:01:27,940 --> 00:01:30,930 We will understand what exactly is going on, build 30 00:01:30,930 --> 00:01:34,240 some intuition, and realize that this is a general 31 00:01:34,240 --> 00:01:37,770 phenomenon that also shows up in many other contexts.