1 00:00:00,610 --> 00:00:02,510 The definition of the Poisson process 2 00:00:02,510 --> 00:00:05,270 gives us information about the probability 3 00:00:05,270 --> 00:00:09,480 that we get k arrivals during an interval of length delta 4 00:00:09,480 --> 00:00:12,570 when delta is a very small number. 5 00:00:12,570 --> 00:00:15,760 How can we find the probability of k arrivals 6 00:00:15,760 --> 00:00:18,590 during an interval of some general length tau, 7 00:00:18,590 --> 00:00:21,630 where tau is no longer a small number? 8 00:00:21,630 --> 00:00:24,700 In particular, we're interested in the random variable denoted 9 00:00:24,700 --> 00:00:27,350 N sub tau, which stands for the number of arrivals 10 00:00:27,350 --> 00:00:29,660 during an interval of length tau. 11 00:00:29,660 --> 00:00:32,940 And we wish to find the PMF of this random variable, 12 00:00:32,940 --> 00:00:36,230 the probability that N sub tau is equal to k, 13 00:00:36,230 --> 00:00:38,190 and which is what we have been denoting 14 00:00:38,190 --> 00:00:41,250 by this particular notation in the context of the Poisson 15 00:00:41,250 --> 00:00:43,360 process. 16 00:00:43,360 --> 00:00:45,910 Now if, instead of the Poisson model, 17 00:00:45,910 --> 00:00:48,250 we had for the Bernoulli process model, 18 00:00:48,250 --> 00:00:49,830 we would know the answer. 19 00:00:49,830 --> 00:00:54,860 S, the number of successes, or number of arrivals in n slots, 20 00:00:54,860 --> 00:00:58,780 has a PMF which is given by the binomial formula. 21 00:00:58,780 --> 00:01:02,900 Can we somehow use what we know about the Bernoulli process 22 00:01:02,900 --> 00:01:06,150 to find the answer for the Poisson process? 23 00:01:06,150 --> 00:01:09,430 The answer is yes, and it involves a limiting argument 24 00:01:09,430 --> 00:01:11,160 of the following kind. 25 00:01:11,160 --> 00:01:14,670 We take the interval from 0 to t and divide it 26 00:01:14,670 --> 00:01:19,230 into a very large number of intervals, so many of them, 27 00:01:19,230 --> 00:01:22,740 where each one of the intervals has a length of delta, 28 00:01:22,740 --> 00:01:26,180 where delta is a small number. 29 00:01:26,180 --> 00:01:29,560 And to push the analogy with the Bernoulli process, 30 00:01:29,560 --> 00:01:35,100 we will be calling those little intervals as slots. 31 00:01:35,100 --> 00:01:40,960 Now during each slot, we may get zero arrivals, one arrival, 32 00:01:40,960 --> 00:01:43,110 but there's also the possibility that there 33 00:01:43,110 --> 00:01:46,979 may be two arrivals, or even more than two arrivals, 34 00:01:46,979 --> 00:01:49,400 happening during one of the slots. 35 00:01:49,400 --> 00:01:52,060 Because of this, the picture that we have here 36 00:01:52,060 --> 00:01:54,850 is not quite the same as for the Bernoulli process 37 00:01:54,850 --> 00:01:57,360 because in the Bernoulli process, each one of the slots 38 00:01:57,360 --> 00:02:00,350 will get only 0 or 1. 39 00:02:00,350 --> 00:02:03,840 So the source of the discrepancy between the two models 40 00:02:03,840 --> 00:02:08,220 is that here, a slot may obtain two or more arrivals. 41 00:02:08,220 --> 00:02:10,250 But how likely is this? 42 00:02:10,250 --> 00:02:14,110 Let us look at the probability that some slot, that 43 00:02:14,110 --> 00:02:19,070 is, any one of the slots, contains two or more arrivals. 44 00:02:19,070 --> 00:02:22,770 That is, we're dealing with the union of the events 45 00:02:22,770 --> 00:02:27,445 that slot i has two or more arrivals. 46 00:02:33,540 --> 00:02:36,250 This event is the union of these events 47 00:02:36,250 --> 00:02:39,810 and, therefore, the probability of this event is less than 48 00:02:39,810 --> 00:02:44,870 or equal than the sum of the probabilities 49 00:02:44,870 --> 00:02:47,829 of the constituents events. 50 00:02:47,829 --> 00:02:50,030 This is an inequality that we have 51 00:02:50,030 --> 00:02:51,730 seen at some point in the past. 52 00:02:51,730 --> 00:02:54,872 And we're calling it to the union bound. 53 00:02:54,872 --> 00:02:57,350 Now what is this summation? 54 00:02:57,350 --> 00:02:59,800 i ranges over the different slots. 55 00:02:59,800 --> 00:03:03,100 And we have tau over delta slots, 56 00:03:03,100 --> 00:03:06,260 so there's so many terms that are being summed. 57 00:03:06,260 --> 00:03:09,100 Now, during any particular slot, the probability 58 00:03:09,100 --> 00:03:13,520 of two or more arrivals is of order delta squared, according 59 00:03:13,520 --> 00:03:15,880 to the definition of the Poisson process. 60 00:03:15,880 --> 00:03:18,800 And this quantity converges to 0 when 61 00:03:18,800 --> 00:03:22,240 we let delta become smaller and smaller. 62 00:03:22,240 --> 00:03:26,230 So this means that the discrepancy between the Poisson 63 00:03:26,230 --> 00:03:30,030 and the Bernoulli model, which was due to the possibility 64 00:03:30,030 --> 00:03:32,910 that we might get two or more arrivals during one 65 00:03:32,910 --> 00:03:36,120 of those slots, this discrepancy is something 66 00:03:36,120 --> 00:03:40,380 that happens with negligible probability. 67 00:03:40,380 --> 00:03:43,610 In other words, the probability that we 68 00:03:43,610 --> 00:03:53,270 get k arrivals in the Poisson model 69 00:03:53,270 --> 00:03:57,540 is approximately the same as the probability 70 00:03:57,540 --> 00:04:01,620 that k slots have an arrival. 71 00:04:08,040 --> 00:04:10,820 Since we're neglecting the possibility 72 00:04:10,820 --> 00:04:13,860 that some slot has two or more arrivals, 73 00:04:13,860 --> 00:04:16,959 this means that the number of arrivals in the Poisson model 74 00:04:16,959 --> 00:04:22,330 will be the same as the number of slots that get an arrival. 75 00:04:22,330 --> 00:04:27,750 This approximate equality becomes more and more exact 76 00:04:27,750 --> 00:04:30,930 as we let delta go to zero. 77 00:04:30,930 --> 00:04:33,510 But now what is this quantity? 78 00:04:33,510 --> 00:04:37,100 The probability that k slots have an arrival 79 00:04:37,100 --> 00:04:39,140 is something that we can calculate 80 00:04:39,140 --> 00:04:41,750 using the binomial probabilities. 81 00:04:41,750 --> 00:04:46,370 Each one of the slots has a certain probability 82 00:04:46,370 --> 00:04:48,290 of having an arrival. 83 00:04:48,290 --> 00:04:53,030 And different slots are independent of each other 84 00:04:53,030 --> 00:04:56,659 by the defining properties of the Poisson process. 85 00:04:56,659 --> 00:04:59,700 Therefore, this approximation that we have developed 86 00:04:59,700 --> 00:05:03,360 satisfies the properties of the Bernoulli process. 87 00:05:03,360 --> 00:05:06,440 We have a certain probability that each slot gets an arrival. 88 00:05:06,440 --> 00:05:09,180 And we have independence across slots. 89 00:05:09,180 --> 00:05:11,540 This means that we can use now the PMF that's 90 00:05:11,540 --> 00:05:14,110 associated with the Bernoulli model 91 00:05:14,110 --> 00:05:18,180 to calculate this quantity and then take the limit, 92 00:05:18,180 --> 00:05:24,020 as delta goes to 0, to obtain a formula for the PMF 93 00:05:24,020 --> 00:05:26,800 for the Poisson process. 94 00:05:26,800 --> 00:05:30,800 In more detail, what we have is the number 95 00:05:30,800 --> 00:05:34,040 of arrivals, which is approximately 96 00:05:34,040 --> 00:05:37,970 the same as the number of slots that have an arrival, obeys 97 00:05:37,970 --> 00:05:41,430 a binomial distribution in the limit as delta 98 00:05:41,430 --> 00:05:44,850 goes to 0-- a binomial distribution in which 99 00:05:44,850 --> 00:05:48,210 the probability of arrival during each one of the slots 100 00:05:48,210 --> 00:05:52,900 is approximately lambda delta and the number of slots 101 00:05:52,900 --> 00:05:54,480 goes to infinity. 102 00:05:54,480 --> 00:05:57,705 And this happens in a way so that the product of the two, 103 00:05:57,705 --> 00:06:01,660 n times p, is equal to-- this term times 104 00:06:01,660 --> 00:06:04,770 this term gives us a lambda times tau. 105 00:06:04,770 --> 00:06:07,120 This term times this term gives us 106 00:06:07,120 --> 00:06:09,250 something that's order of delta. 107 00:06:09,250 --> 00:06:11,130 So it's negligible. 108 00:06:11,130 --> 00:06:14,560 So we have this equality, and so this 109 00:06:14,560 --> 00:06:18,210 is approximately lambda tau with the approximation 110 00:06:18,210 --> 00:06:23,310 becoming more and more exact as we let delta go to zero. 111 00:06:23,310 --> 00:06:28,130 So all we need to do is to take the formula for the Bernoulli 112 00:06:28,130 --> 00:06:29,280 process. 113 00:06:29,280 --> 00:06:34,510 Use these values of p and n and take the limit. 114 00:06:34,510 --> 00:06:37,659 But this is a problem that we have already encountered 115 00:06:37,659 --> 00:06:39,230 and have analyzed. 116 00:06:39,230 --> 00:06:41,810 If we let n go to infinity, p goes to 0 117 00:06:41,810 --> 00:06:44,030 so that their product stays constant, 118 00:06:44,030 --> 00:06:49,040 we have shown that the binomial PMF converges 119 00:06:49,040 --> 00:06:52,710 to the so-called Poisson PMF that takes this form. 120 00:06:52,710 --> 00:06:54,960 Notice one small difference-- n times 121 00:06:54,960 --> 00:06:58,450 p here is equal to lambda, whereas here, n times 122 00:06:58,450 --> 00:06:59,780 p is equal to lambda t. 123 00:06:59,780 --> 00:07:02,240 This means that we need to apply this formula, 124 00:07:02,240 --> 00:07:05,210 but with lambda replaced by lambda t, 125 00:07:05,210 --> 00:07:08,090 and this gives us the final answer. 126 00:07:08,090 --> 00:07:12,320 This is the probability of k arrivals during a time interval 127 00:07:12,320 --> 00:07:15,060 of lenght t in the Poisson process. 128 00:07:15,060 --> 00:07:22,410 And this is a so-called Poisson PMF with parameter lambda tau. 129 00:07:22,410 --> 00:07:27,780 To summarize, our strategy was to argue that the Poisson 130 00:07:27,780 --> 00:07:32,870 process is increasingly accurately described 131 00:07:32,870 --> 00:07:35,860 by a Bernoulli process if we discretize 132 00:07:35,860 --> 00:07:38,740 time in a very fine discretization. 133 00:07:38,740 --> 00:07:41,350 And the approximation becomes exact in the limit 134 00:07:41,350 --> 00:07:43,840 when the discretization is very fine. 135 00:07:43,840 --> 00:07:47,040 So we took the corresponding binomial formula 136 00:07:47,040 --> 00:07:49,980 for the Bernoulli process and took the limit 137 00:07:49,980 --> 00:07:52,290 to that's associated with the parameters 138 00:07:52,290 --> 00:07:56,580 that we would obtain if we have a very fine discretization. 139 00:07:56,580 --> 00:07:59,567 And this gave us the final formula.