1 00:00:00,150 --> 00:00:03,240 Back, I want to talk a little more about this idea 2 00:00:03,240 --> 00:00:04,950 that I introduced in the last video, 3 00:00:04,950 --> 00:00:09,210 which is the fact that whenever we have mechanisms 4 00:00:09,210 --> 00:00:11,610 or events governed by fixed rate 5 00:00:11,610 --> 00:00:14,430 that occur regardless of what happened before, 6 00:00:14,430 --> 00:00:17,974 we call that a Poisson process in both discrete time 7 00:00:17,974 --> 00:00:19,560 or in the limit of very small discrete steps, 8 00:00:19,560 --> 00:00:20,673 continuous time. 9 00:00:22,740 --> 00:00:24,540 And really I wanted to talk a little more 10 00:00:24,540 --> 00:00:26,760 about the mathematics of these processes 11 00:00:26,760 --> 00:00:29,700 because we encountered two problem in trying to code up 12 00:00:29,700 --> 00:00:33,840 a very simple random or stochastic simulation 13 00:00:33,840 --> 00:00:35,700 of a simple model. 14 00:00:35,700 --> 00:00:38,880 So one of those is that what happens is if those rates 15 00:00:38,880 --> 00:00:39,903 are very high? 16 00:00:40,739 --> 00:00:43,680 How do we make it so that the probability 17 00:00:43,680 --> 00:00:47,733 in any one step of something occurring is smaller than one? 18 00:00:48,840 --> 00:00:52,740 The other one is if those probabilities are very high, 19 00:00:52,740 --> 00:00:54,718 you can have multiple events happening at the same time, 20 00:00:54,718 --> 00:00:56,013 and you don't want that. 21 00:00:56,957 --> 00:00:59,520 I don't wanna both infect and recover, 22 00:00:59,520 --> 00:01:01,980 infect someone with a disease and recover from the disease 23 00:01:01,980 --> 00:01:03,030 at the same time. 24 00:01:03,030 --> 00:01:08,030 Really it's, the model, the system that we're thinking of 25 00:01:08,100 --> 00:01:09,840 doesn't have those simultaneous event. 26 00:01:09,840 --> 00:01:12,120 One has to happen before the other. 27 00:01:12,120 --> 00:01:15,093 So we don't want our random simulation to do that. 28 00:01:16,470 --> 00:01:18,060 So we want to think about what happens, 29 00:01:18,060 --> 00:01:21,093 how do we deal with competing Poisson processes? 30 00:01:23,943 --> 00:01:27,527 Okay, so here's how to think about it, right? 31 00:01:30,600 --> 00:01:33,720 Let's say we're only dealing with people leaving, 32 00:01:33,720 --> 00:01:36,960 so our departure term from the cloning factory, right? 33 00:01:36,960 --> 00:01:39,960 Well, what is the probability that they leave 34 00:01:39,960 --> 00:01:41,400 in the first step? 35 00:01:41,400 --> 00:01:45,933 So someone in the clone factory. 36 00:01:50,730 --> 00:01:51,563 Whoops. 37 00:01:53,970 --> 00:01:55,840 In the clone factory 38 00:02:00,960 --> 00:02:05,670 leaving in first step. 39 00:02:10,110 --> 00:02:11,820 Well that's pretty simple, right? 40 00:02:11,820 --> 00:02:16,410 So if I say that I have a departure rate or a leaving rate, 41 00:02:16,410 --> 00:02:20,460 which we call l, and an increment of time 42 00:02:20,460 --> 00:02:22,773 that we call t, delta t, 43 00:02:24,660 --> 00:02:27,740 we said we simply test l times delta t 44 00:02:27,740 --> 00:02:30,540 if is my random number smaller than that. 45 00:02:30,540 --> 00:02:33,540 So the probability that I leave in the first step 46 00:02:33,540 --> 00:02:36,543 should be l times delta t. 47 00:02:40,680 --> 00:02:44,080 The probability that I leave in second step 48 00:02:48,390 --> 00:02:52,260 would simply be me not leaving in the first step, 49 00:02:52,260 --> 00:02:53,943 one minus l delta t, 50 00:02:55,500 --> 00:02:57,750 then leaving in the second one. 51 00:02:57,750 --> 00:03:01,500 So the first time, first day, first step in the process, 52 00:03:01,500 --> 00:03:03,780 you go over me, you ask me, do you choose to leave? 53 00:03:03,780 --> 00:03:05,670 Roll the dice, no. 54 00:03:05,670 --> 00:03:06,930 And then in the second step. 55 00:03:06,930 --> 00:03:09,270 Do you choose to leave roll the dice? Yes. 56 00:03:09,270 --> 00:03:12,390 I run a random number smaller than l times delta t, 57 00:03:12,390 --> 00:03:13,983 I choose to leave, okay. 58 00:03:14,850 --> 00:03:16,680 Well here, what I want to talk about, 59 00:03:16,680 --> 00:03:18,660 the reason I'm going through this is I want to talk about 60 00:03:18,660 --> 00:03:20,850 what happens if we renormalize rates, right? 61 00:03:20,850 --> 00:03:24,090 So if l times delta t is too big, let's say it's 1.2, 62 00:03:24,090 --> 00:03:26,880 everyone's gonna leave and it doesn't matter if it's 1.2 63 00:03:26,880 --> 00:03:30,780 or 1.5, people are not leaving faster, 64 00:03:30,780 --> 00:03:32,643 they're limited by this delta t. 65 00:03:33,570 --> 00:03:35,673 So I could renormalize, right? 66 00:03:37,110 --> 00:03:40,650 By that I just mean, you know, 67 00:03:40,650 --> 00:03:43,170 sort of divide this delta t by something. 68 00:03:43,170 --> 00:03:46,320 So let's say that we now have delta t tilda, 69 00:03:46,320 --> 00:03:49,770 which I'm just gonna call delta t over two. 70 00:03:49,770 --> 00:03:53,220 So instead of days we're thinking of half days, 71 00:03:53,220 --> 00:03:55,740 or instead of hours, we're thinking of half hours. 72 00:03:55,740 --> 00:03:57,243 That's sort of the process. 73 00:03:58,980 --> 00:04:03,930 Okay, well, what I wanna see is if I renormalize like that, 74 00:04:03,930 --> 00:04:06,370 how much of a difference does it really make 75 00:04:07,830 --> 00:04:10,590 between this renormalize stochastic simulation 76 00:04:10,590 --> 00:04:15,090 and the previous stochastic simulation with delta t? 77 00:04:15,090 --> 00:04:19,233 Well, now the probability of leaving in the first delta t, 78 00:04:24,120 --> 00:04:26,260 so leaving in first delta t 79 00:04:30,810 --> 00:04:35,760 is, well, I either leave in the very first delta t over two, 80 00:04:35,760 --> 00:04:37,863 which is what I'm testing over now. 81 00:04:40,320 --> 00:04:42,870 So if I do that, that's good. 82 00:04:42,870 --> 00:04:47,870 Or I choose not to leave in the first delta t over two, 83 00:04:48,030 --> 00:04:52,420 and I leave in the second 84 00:04:53,610 --> 00:04:54,783 delta t over two. 85 00:04:56,370 --> 00:05:00,453 By the way, I said I anthropomorphize a lot. 86 00:05:03,000 --> 00:05:05,970 Whenever I say I choose to leave or I choose not to leave, 87 00:05:05,970 --> 00:05:07,050 that's just wrong, right? 88 00:05:07,050 --> 00:05:09,270 It's just a pure roll of the dice. 89 00:05:09,270 --> 00:05:12,600 Does the gut of the dice 90 00:05:12,600 --> 00:05:15,060 choose to make that person leave or not, 91 00:05:15,060 --> 00:05:17,553 the random number generator being the gut here? 92 00:05:18,750 --> 00:05:20,733 Okay, so this is what I have. 93 00:05:21,960 --> 00:05:26,580 Well, I'm gonna have this l times delta two over two here, 94 00:05:26,580 --> 00:05:29,640 another l times t delta two over here, 95 00:05:29,640 --> 00:05:31,110 and then some other term. 96 00:05:31,110 --> 00:05:33,880 So really what I have is l times delta t 97 00:05:35,430 --> 00:05:40,430 minus l delta t over two squared. 98 00:05:41,160 --> 00:05:45,270 Okay, so I'm pretty close to what I had originally now, 99 00:05:45,270 --> 00:05:48,210 just with a correction factor here. 100 00:05:48,210 --> 00:05:49,043 Interesting. 101 00:05:50,850 --> 00:05:53,913 With the probability of leaving in second step, 102 00:05:57,490 --> 00:05:59,477 in the second delta t, right? 103 00:06:04,590 --> 00:06:06,740 Well, that one's a little more complicated. 104 00:06:08,040 --> 00:06:10,320 So regardless, I need to not leave 105 00:06:10,320 --> 00:06:12,990 in the first two delta t over two, 106 00:06:12,990 --> 00:06:16,860 that's one minus l delta t over two. 107 00:06:16,860 --> 00:06:20,193 So I don't leave in the first two half day or half hours. 108 00:06:21,810 --> 00:06:24,033 And then I could leave in the third one. 109 00:06:27,090 --> 00:06:30,153 Or I don't leave in the first three, 110 00:06:33,450 --> 00:06:38,450 and I leave in the fourth one, right? 111 00:06:38,580 --> 00:06:41,130 If you're getting confused here, just think of it this way. 112 00:06:41,130 --> 00:06:45,400 The first calculation we did way over here at the beginning 113 00:06:46,620 --> 00:06:49,050 was looking at whether people were leaving in the first 114 00:06:49,050 --> 00:06:53,310 or second step where step is in steps of delta t. 115 00:06:53,310 --> 00:06:55,863 So either left initially or in the second one. 116 00:06:57,870 --> 00:07:00,470 What we're doing now after the renormalization 117 00:07:00,470 --> 00:07:02,043 is just testing more. 118 00:07:03,450 --> 00:07:05,820 So you can leave in the first delta two over two 119 00:07:05,820 --> 00:07:08,190 or the second one, which both mean leaving 120 00:07:08,190 --> 00:07:11,010 in the first delta t, or you can leave in the third delta t 121 00:07:11,010 --> 00:07:12,450 over two or the fourth one, 122 00:07:12,450 --> 00:07:16,323 which both mean leaving in the second delta t, okay? 123 00:07:18,240 --> 00:07:20,640 I think I erased something important here. Oops. 124 00:07:21,840 --> 00:07:23,823 Over two, okay. 125 00:07:24,930 --> 00:07:29,133 Well, if you do the math of like expending all of these, 126 00:07:31,230 --> 00:07:33,250 essentially what you're gonna get here 127 00:07:41,910 --> 00:07:43,290 is something pretty simple. 128 00:07:43,290 --> 00:07:47,853 So you're gonna get, trying to do it quickly. 129 00:07:49,530 --> 00:07:53,190 You're gonna get, now then the t over two from this, 130 00:07:53,190 --> 00:07:54,753 you're gonna get two of those. 131 00:07:56,460 --> 00:07:58,500 So you're gonna get your l delta t, 132 00:07:58,500 --> 00:08:01,083 which is gonna be the same as this term here. 133 00:08:04,290 --> 00:08:05,883 You're gonna get your, 134 00:08:07,717 --> 00:08:10,337 you're gonna get two l delta t squared 135 00:08:11,910 --> 00:08:14,493 over two where the two is in squared. 136 00:08:18,330 --> 00:08:20,380 And that's gonna be again your term here. 137 00:08:21,990 --> 00:08:24,180 And then you're gonna get in the same thing. 138 00:08:24,180 --> 00:08:28,143 You're gonna get some some third and fourth order term, 139 00:08:29,820 --> 00:08:33,303 and by that I mean proportional to delta t over three. 140 00:08:35,340 --> 00:08:36,990 Okay, so it's sort of the same thing. 141 00:08:36,990 --> 00:08:39,840 We're still getting the same probability 142 00:08:39,840 --> 00:08:42,270 to first approximation of getting out 143 00:08:42,270 --> 00:08:43,770 in the first or second step, 144 00:08:43,770 --> 00:08:46,320 and then we have some order-order comparison. 145 00:08:46,320 --> 00:08:49,290 What this means is that if our delta t are pretty big, 146 00:08:49,290 --> 00:08:52,320 if our probabilities l times delta t, 147 00:08:52,320 --> 00:08:53,820 maybe I should keep an l here, 148 00:08:54,750 --> 00:08:58,980 are pretty big, whenever we renormalize our rates, 149 00:08:58,980 --> 00:09:01,590 we're not gonna get exactly the same dynamics, right? 150 00:09:01,590 --> 00:09:04,680 We're gonna get some correction terms all over. 151 00:09:04,680 --> 00:09:07,290 But if our l times delta t gets really, really small, 152 00:09:07,290 --> 00:09:10,173 imagine that l times delta t is one in 1,000. 153 00:09:11,430 --> 00:09:14,763 Well, this l times delta t cubed, you know, 154 00:09:15,852 --> 00:09:17,700 goes from 10 to the minus three to 10 to the minus nine. 155 00:09:17,700 --> 00:09:19,290 And that's tiny, right? 156 00:09:19,290 --> 00:09:21,300 So the smaller our rates are, 157 00:09:21,300 --> 00:09:24,330 eventually those error terms don't matter anymore, 158 00:09:24,330 --> 00:09:27,300 and renormalizing your rates don't matter. 159 00:09:27,300 --> 00:09:29,610 So one easy way to like solve this problem 160 00:09:29,610 --> 00:09:32,460 is just to use small discrete increment, 161 00:09:32,460 --> 00:09:34,140 and then you're free. 162 00:09:34,140 --> 00:09:36,060 You know, if you make sure that all your rates 163 00:09:36,060 --> 00:09:38,940 are at most of the order of one in 1,000, 164 00:09:38,940 --> 00:09:41,130 all your your rates time delta t, 165 00:09:41,130 --> 00:09:43,650 then you're free to renormalize sort of as you please, 166 00:09:43,650 --> 00:09:46,080 and you know that the errors that you make 167 00:09:46,080 --> 00:09:47,883 are gonna be tiny, right? 168 00:09:49,500 --> 00:09:51,750 So that's one way to get around the problem 169 00:09:51,750 --> 00:09:54,360 of what if my rates are really big, just renormalize. 170 00:09:54,360 --> 00:09:57,780 You were thinking in terms of discrete steps that were days, 171 00:09:57,780 --> 00:10:00,330 think in terms of discrete step that are minutes, 172 00:10:00,330 --> 00:10:01,770 and now it's gonna be pretty smooth 173 00:10:01,770 --> 00:10:03,960 and you're not gonna have any problem. 174 00:10:03,960 --> 00:10:05,493 So that's one easy thing. 175 00:10:06,870 --> 00:10:09,450 The more interesting thing, and here I'm gonna go 176 00:10:09,450 --> 00:10:12,330 a little deeper in the math, is this idea of 177 00:10:12,330 --> 00:10:14,403 what if I can do two things, right? 178 00:10:15,810 --> 00:10:18,960 So you might have like my face over this. 179 00:10:18,960 --> 00:10:23,960 So what if I have competing processes? 180 00:10:24,000 --> 00:10:28,080 Which is what we had in the cloning building, 181 00:10:28,080 --> 00:10:31,560 in the clone factory because people could both leave 182 00:10:31,560 --> 00:10:33,240 or get cloned, and the question is 183 00:10:33,240 --> 00:10:35,100 which one do I do first, right? 184 00:10:35,100 --> 00:10:37,113 So here what we want to talk about is, 185 00:10:38,256 --> 00:10:39,810 let's focus again on just one person 186 00:10:39,810 --> 00:10:41,220 and let that person be me. 187 00:10:41,220 --> 00:10:44,190 Okay, so I entered a clone factory, then the question is, 188 00:10:44,190 --> 00:10:47,373 at every delta t I'm gonna be asked, do you choose to leave? 189 00:10:48,240 --> 00:10:51,450 And I can only get cloned while I'm in the building. 190 00:10:51,450 --> 00:10:54,640 So really what I want to know beforehand is 191 00:10:55,680 --> 00:10:58,260 how long am I gonna stay in this building, right? 192 00:10:58,260 --> 00:11:00,813 So what is the period of time, 193 00:11:03,330 --> 00:11:06,570 let's call it l, so the duration l for length, 194 00:11:06,570 --> 00:11:10,293 length of time that someone, 195 00:11:12,300 --> 00:11:15,183 and that's me, stays in the building. 196 00:11:19,800 --> 00:11:20,633 Okay. 197 00:11:22,260 --> 00:11:26,010 Well, the easiest way to derive this for Poisson processes, 198 00:11:26,010 --> 00:11:28,740 or the way that's most intuitive to me 199 00:11:28,740 --> 00:11:30,903 is to go through the cumulative, 200 00:11:32,790 --> 00:11:34,350 I'll spell it out for this one, 201 00:11:34,350 --> 00:11:39,123 cumulative distribution function. 202 00:11:43,680 --> 00:11:46,500 I wasn't even sure saying the word. 203 00:11:46,500 --> 00:11:49,020 It's usually just called CDF, right? 204 00:11:49,020 --> 00:11:53,400 So this is a function as a function of time, 205 00:11:53,400 --> 00:11:57,450 let's say l that starts at zero. 206 00:11:57,450 --> 00:12:00,360 So there's a probability zero that you don't stay 207 00:12:00,360 --> 00:12:02,160 in the building at all because I'm sort of saying 208 00:12:02,160 --> 00:12:03,900 I'm in the building. 209 00:12:03,900 --> 00:12:07,470 And this is gonna go up in some way, 210 00:12:07,470 --> 00:12:08,820 but can't go beyond one 211 00:12:08,820 --> 00:12:12,090 'cause it's just a sum of probabilities. 212 00:12:12,090 --> 00:12:14,940 So essentially here what I'm calculating 213 00:12:14,940 --> 00:12:18,003 is what's the probability that I've left. 214 00:12:20,400 --> 00:12:22,200 And I shouldn't use the same l here. 215 00:12:24,489 --> 00:12:26,580 Let's just call it t. 216 00:12:26,580 --> 00:12:29,610 So it's the probability that I've left before time t, 217 00:12:29,610 --> 00:12:33,510 so it's zero at t equals zero, and it initially goes up, 218 00:12:33,510 --> 00:12:34,920 and eventually it's gonna be one. 219 00:12:34,920 --> 00:12:36,480 What's the probability that someone 220 00:12:36,480 --> 00:12:38,640 has left before one million years? 221 00:12:38,640 --> 00:12:39,473 It's one, right? 222 00:12:39,473 --> 00:12:41,800 Nobody's staying in the building for that long. 223 00:12:43,470 --> 00:12:46,490 So what does that look like? So let's call it... 224 00:12:48,360 --> 00:12:50,790 I'm just gonna use l for all like things. 225 00:12:50,790 --> 00:12:53,553 So hopefully it doesn't get too confusing. 226 00:12:59,370 --> 00:13:04,290 So here, to be able to do math, 227 00:13:04,290 --> 00:13:05,670 we're gonna have to take the limit 228 00:13:05,670 --> 00:13:07,800 of very small time step delta t. 229 00:13:07,800 --> 00:13:12,800 Like I said when we were talking about renormalization, 230 00:13:12,810 --> 00:13:14,820 as delta t goes to zero, 231 00:13:14,820 --> 00:13:17,700 then the scale of our discrete process doesn't matter. 232 00:13:17,700 --> 00:13:19,560 It doesn't matter because now we have essentially 233 00:13:19,560 --> 00:13:21,360 a continuous time dynamics, 234 00:13:21,360 --> 00:13:23,710 and it makes it a lot easier to do math, right? 235 00:13:30,240 --> 00:13:33,390 And I'm gonna explain what I'm writing in a second, okay? 236 00:13:33,390 --> 00:13:38,390 So the probability that I've left in a time smaller than t, 237 00:13:39,360 --> 00:13:42,300 than l, sorry, the probability that I've left 238 00:13:42,300 --> 00:13:45,753 in a time smaller than l, which is what this mean, 239 00:13:46,890 --> 00:13:50,940 well, it's everything but people that are still there. 240 00:13:50,940 --> 00:13:53,580 So whenever you say everything but in probability, 241 00:13:53,580 --> 00:13:55,140 that means one minus, 242 00:13:55,140 --> 00:13:58,470 everything but the people that are still there. 243 00:13:58,470 --> 00:14:01,140 And the people that are still there 244 00:14:01,140 --> 00:14:02,970 through all the time steps, 245 00:14:02,970 --> 00:14:04,980 and how many time steps have we taken? 246 00:14:04,980 --> 00:14:07,413 It's l divided by delta t, right? 247 00:14:08,550 --> 00:14:13,550 So this here is number of steps that we've done. 248 00:14:14,430 --> 00:14:17,940 So if l is a million year and delta t is a year 249 00:14:17,940 --> 00:14:19,890 to make it easy, then we have, you know, 250 00:14:19,890 --> 00:14:21,570 a million a year divided by a year, 251 00:14:21,570 --> 00:14:23,673 we have one million times steps, right? 252 00:14:25,890 --> 00:14:28,500 And then for all of those times steps 253 00:14:28,500 --> 00:14:31,170 this one minus l times delta t 254 00:14:31,170 --> 00:14:34,530 is the probability of not leaving. 255 00:14:34,530 --> 00:14:36,810 So all that's left is everything 256 00:14:36,810 --> 00:14:38,985 but the people that have not left in any 257 00:14:38,985 --> 00:14:43,533 of these l divided by delta t times steps. 258 00:14:46,140 --> 00:14:49,560 Cool, cool, cool. Well, what is this? 259 00:14:49,560 --> 00:14:54,560 This is equal to one minus the exponential of minus lL. 260 00:14:56,070 --> 00:14:59,523 And if this feels like it's out of nowhere right now, 261 00:15:01,470 --> 00:15:04,950 I'll invite you to watch a bonus video that I'll be making. 262 00:15:04,950 --> 00:15:06,783 Let me just highlight this. 263 00:15:08,070 --> 00:15:11,070 So there's gonna be a bonus video 264 00:15:11,070 --> 00:15:15,550 on Taylor series and approximation 265 00:15:16,800 --> 00:15:18,090 to explain this. 266 00:15:18,090 --> 00:15:21,780 But really is that when we have a very small probability, 267 00:15:21,780 --> 00:15:24,270 and we know it's small because delta t 268 00:15:24,270 --> 00:15:29,090 is going to zero here, we can rewrite that as... 269 00:15:30,780 --> 00:15:33,750 Well, I don't wanna go through the video right now. 270 00:15:33,750 --> 00:15:34,583 I wanna keep it short. 271 00:15:34,583 --> 00:15:36,400 I've already been talking for a while, 272 00:15:37,350 --> 00:15:38,463 so sorry about that. 273 00:15:39,664 --> 00:15:42,180 It basically involves the fact that delta t 274 00:15:42,180 --> 00:15:44,670 is gonna be really small so that we know this is gonna be 275 00:15:44,670 --> 00:15:46,470 like an exponential distribution, okay, 276 00:15:46,470 --> 00:15:47,670 an exponential function. 277 00:15:47,670 --> 00:15:49,173 So we have this one minus, 278 00:15:51,630 --> 00:15:54,030 one minus exponential of minus small l, 279 00:15:54,030 --> 00:15:55,800 the rate times arbitration, okay? 280 00:15:55,800 --> 00:15:58,230 So that's our cumulative distribution function. 281 00:15:58,230 --> 00:16:00,000 This cumulative distribution function, 282 00:16:00,000 --> 00:16:01,860 it's called cumulative because it's the sum 283 00:16:01,860 --> 00:16:05,940 of the probability of, I'm gonna try and highlight here, 284 00:16:05,940 --> 00:16:08,550 of leaving at zero, leaving it at the first step, 285 00:16:08,550 --> 00:16:09,570 second step, third step. 286 00:16:09,570 --> 00:16:12,270 And you sum them up to ask who has left 287 00:16:12,270 --> 00:16:14,040 at any time before this. 288 00:16:14,040 --> 00:16:16,533 So you're just summing all of the steps, right? 289 00:16:18,090 --> 00:16:19,773 So in continuous really, 290 00:16:22,184 --> 00:16:25,493 this c of l is the integral 291 00:16:27,390 --> 00:16:31,290 of the probability of leaving at a time exactly l, 292 00:16:31,290 --> 00:16:34,803 well this is getting confusing with all the l's. 293 00:16:36,870 --> 00:16:40,413 It's at a time exactly t overall possible time t. 294 00:16:41,820 --> 00:16:44,430 So the probability that I leave, 295 00:16:44,430 --> 00:16:48,210 and that's what I'm calling P at a time exactly l 296 00:16:48,210 --> 00:16:52,030 is the derivative of cL 297 00:16:53,640 --> 00:16:57,320 over dL, right? 298 00:16:57,320 --> 00:17:01,020 So if the cumulative is the sum of all the possibilities 299 00:17:01,020 --> 00:17:03,780 of leaving at all events, it's the integral, 300 00:17:03,780 --> 00:17:05,940 well to reverse that relationship, 301 00:17:05,940 --> 00:17:08,193 the probability distribution function, 302 00:17:09,510 --> 00:17:12,510 P of L, is simply gonna be the derivative. 303 00:17:12,510 --> 00:17:17,340 Well then the derivative of this one minus exponential 304 00:17:17,340 --> 00:17:19,770 of minus l times l is is really simple. 305 00:17:19,770 --> 00:17:23,763 It's simply l e the minus lL. 306 00:17:25,620 --> 00:17:28,230 Well you might have seen those distribution before. 307 00:17:28,230 --> 00:17:32,357 Usually there is simply t or x, 308 00:17:33,810 --> 00:17:35,520 or whatever variable you want. 309 00:17:35,520 --> 00:17:38,940 And this lambda is the rate of these Poisson process. 310 00:17:38,940 --> 00:17:40,710 This is a Poisson distribution, right? 311 00:17:40,710 --> 00:17:43,620 So it all sort of ties together. 312 00:17:43,620 --> 00:17:46,080 We know that the average is one over lambda. 313 00:17:46,080 --> 00:17:47,760 So the average time that it takes to leave 314 00:17:47,760 --> 00:17:50,100 is one over l in this case, right? 315 00:17:50,100 --> 00:17:55,100 So if your rate at which you leave is 0.1 per week, 316 00:17:55,200 --> 00:17:57,510 we know that it's gonna take you on average 10 weeks 317 00:17:57,510 --> 00:17:59,523 to leave the cloning factory, right? 318 00:18:00,810 --> 00:18:02,433 Very mysterious building. 319 00:18:04,440 --> 00:18:08,400 Okay, well, so now we're curious about competing processes. 320 00:18:08,400 --> 00:18:10,120 So what is the probability 321 00:18:15,810 --> 00:18:20,810 of cloning before leaving? 322 00:18:25,560 --> 00:18:29,610 Well, what we're gonna wanna do is integrate 323 00:18:29,610 --> 00:18:34,023 over the probability of leaving at time exactly l. 324 00:18:37,050 --> 00:18:40,200 And then if I know that I'm leaving at time l, 325 00:18:40,200 --> 00:18:43,110 I wanna ask what's the probability that I've cloned myself 326 00:18:43,110 --> 00:18:45,063 in a time less than l. 327 00:18:47,580 --> 00:18:50,430 Well this probability of cloning myself is gonna be 328 00:18:50,430 --> 00:18:52,920 the same cumulative distribution here, 329 00:18:52,920 --> 00:18:54,810 it's gonna be the same as this 330 00:18:54,810 --> 00:18:57,993 but with a rate c instead of l, right? 331 00:18:59,160 --> 00:19:04,160 So it's gonna be one minus e 332 00:19:04,740 --> 00:19:08,970 to the minus cL is the probability that I've cloned myself 333 00:19:08,970 --> 00:19:11,320 before l, so it's this cumulative distribution. 334 00:19:13,470 --> 00:19:15,780 And then I'm just integrating those 335 00:19:15,780 --> 00:19:18,180 over all possible times to l. 336 00:19:18,180 --> 00:19:20,640 All right, I'm running out space here, let me switch. 337 00:19:20,640 --> 00:19:24,160 So very quickly, my probability I was integrating 338 00:19:25,980 --> 00:19:29,400 from zero to infinity, the probability that I leave 339 00:19:29,400 --> 00:19:32,370 at time l which is a Poisson distribution, 340 00:19:32,370 --> 00:19:35,193 I can write that from memory. 341 00:19:36,750 --> 00:19:39,330 And then the probability that I've cloned myself 342 00:19:39,330 --> 00:19:44,103 before this time l you said is one minus e cL. 343 00:19:48,150 --> 00:19:48,983 Okay. 344 00:19:53,850 --> 00:19:58,850 So if you draw on your own calculus days, 345 00:20:03,600 --> 00:20:08,073 you just like split up this integral into two terms. 346 00:20:12,840 --> 00:20:17,310 And now we have this fun sum of rates here, 347 00:20:17,310 --> 00:20:21,993 c plus l when I do the product of my two exponential terms. 348 00:20:24,570 --> 00:20:25,403 Okay? 349 00:20:27,330 --> 00:20:31,260 Well this one is an easy integral 350 00:20:31,260 --> 00:20:33,360 if you remember all your rules. 351 00:20:33,360 --> 00:20:36,630 So that that's simply, you know, this l e minus lL 352 00:20:36,630 --> 00:20:41,630 is really the derivative of the exponential 353 00:20:42,780 --> 00:20:47,010 minus lL and it's minus that, right? 354 00:20:47,010 --> 00:20:50,823 And I'm evaluating it at infinity and at zero. 355 00:20:54,030 --> 00:20:56,493 So from this I'm just gonna get, 356 00:20:58,770 --> 00:21:02,223 it's minus zero minus minus one, so it's one. 357 00:21:04,710 --> 00:21:09,710 Okay, this one is gonna give me l over c plus l. 358 00:21:09,990 --> 00:21:12,723 I'm just taking the integral of the exponential. 359 00:21:18,060 --> 00:21:21,450 And I'm, again, evaluating, this is my notation by the way, 360 00:21:21,450 --> 00:21:23,430 for like evaluating at l equal infinity. 361 00:21:23,430 --> 00:21:26,763 Those are my boundaries on my definite integral. 362 00:21:28,500 --> 00:21:30,200 So this one is just gonna give me, 363 00:21:33,510 --> 00:21:36,423 just gonna give me l over c plus l. 364 00:21:37,380 --> 00:21:41,970 So I have one minus l over c plus l. 365 00:21:41,970 --> 00:21:44,190 I'm just gonna put that on the same denominator, 366 00:21:44,190 --> 00:21:48,363 so I get c plus l minus l over c plus l. 367 00:21:50,970 --> 00:21:55,113 And then I get c over c plus l. 368 00:21:57,660 --> 00:21:59,910 Okay, well you might have a little bit, 369 00:21:59,910 --> 00:22:02,820 you might either be bored, or if you followed it all, 370 00:22:02,820 --> 00:22:06,330 you might have a little bit of a eureka moment here. 371 00:22:06,330 --> 00:22:08,133 Let me, like, erase some of this. 372 00:22:10,650 --> 00:22:11,973 What is this really? 373 00:22:14,220 --> 00:22:18,990 Well, when we took the product here 374 00:22:18,990 --> 00:22:23,990 of two of our Poisson processes, 375 00:22:24,900 --> 00:22:28,290 well, we just learned something here with this one, 376 00:22:28,290 --> 00:22:33,290 is that the rate 377 00:22:35,880 --> 00:22:37,060 of a sum 378 00:22:41,435 --> 00:22:43,102 of Poisson processes 379 00:22:47,250 --> 00:22:51,750 is the sum of the rates. 380 00:22:51,750 --> 00:22:54,290 Well really a sum of Poisson processes is a, 381 00:22:58,364 --> 00:23:00,930 is simply, I mean, is a Poisson process 382 00:23:00,930 --> 00:23:02,583 with the sum of the rates. 383 00:23:04,170 --> 00:23:07,440 Okay, so that's interesting. So now what is this saying? 384 00:23:07,440 --> 00:23:10,320 Is saying that one way to do our previous code 385 00:23:10,320 --> 00:23:12,427 is not to go over everyone and ask, 386 00:23:12,427 --> 00:23:16,320 "Do you leave, do you clone yourself?" 387 00:23:16,320 --> 00:23:18,600 We can combine those two Poisson process 388 00:23:18,600 --> 00:23:21,690 and ask every person, "Do you do something?" 389 00:23:21,690 --> 00:23:24,483 And doing something happens at a rate c plus l. 390 00:23:25,890 --> 00:23:29,730 Okay, and then when they do something, 391 00:23:29,730 --> 00:23:31,890 we can ask, "Well, what's the probability 392 00:23:31,890 --> 00:23:34,767 that this something was a cloning event?" 393 00:23:36,960 --> 00:23:41,200 Well, the probability 394 00:23:45,750 --> 00:23:48,280 of Poisson process, 395 00:23:51,300 --> 00:23:53,220 and I'm kind of gonna call it lambda one. 396 00:23:53,220 --> 00:23:55,450 So a Poisson process which rate lambda one 397 00:23:57,300 --> 00:24:00,543 occurring, and there's probably gonna be a typo here. 398 00:24:01,380 --> 00:24:03,300 I apologize, 399 00:24:03,300 --> 00:24:08,300 before a Poisson process 400 00:24:09,180 --> 00:24:11,080 with a rate lambda two 401 00:24:13,500 --> 00:24:16,650 is simply gonna be lambda one over the sum 402 00:24:16,650 --> 00:24:18,663 of lambda one plus lambda two. 403 00:24:21,270 --> 00:24:24,150 That's what we calculated really. 404 00:24:24,150 --> 00:24:27,120 And this is an incredibly useful result. 405 00:24:27,120 --> 00:24:30,270 I use it, you know, all the time in actual paper. 406 00:24:30,270 --> 00:24:31,980 One way to think about this is, you know, 407 00:24:31,980 --> 00:24:36,033 imagine you're a grad student, you're in Burlington, 408 00:24:36,900 --> 00:24:40,920 your parents are far away, and they all call you. 409 00:24:40,920 --> 00:24:43,650 Let's say that your mom calls you on average once a week 410 00:24:43,650 --> 00:24:46,323 and your dad calls you on average twice a week. 411 00:24:47,460 --> 00:24:51,090 Well that means you get three phone calls a week, right? 412 00:24:51,090 --> 00:24:53,430 So this is this conclusion in blue, 413 00:24:53,430 --> 00:24:55,320 the rate of a sum of Poisson processes 414 00:24:55,320 --> 00:24:57,150 is the sum of the rates. 415 00:24:57,150 --> 00:25:00,450 So if you have one phone call from your mom a week, 416 00:25:00,450 --> 00:25:03,840 two from your dad a week, you get three phone calls a week. 417 00:25:03,840 --> 00:25:07,830 Easy. Now, let's say that right now your phone rings. 418 00:25:07,830 --> 00:25:11,520 What's the probability that it's your dad, right? 419 00:25:11,520 --> 00:25:14,880 Well he calls, like, two out of three calls, 420 00:25:14,880 --> 00:25:16,227 he calls twice a week, 421 00:25:16,227 --> 00:25:19,020 and you get three phone calls a week. 422 00:25:19,020 --> 00:25:21,660 So two out of three calls are from your dad. 423 00:25:21,660 --> 00:25:24,180 So it's this two over two plus one. 424 00:25:24,180 --> 00:25:25,130 It's just this 2/3. 425 00:25:27,060 --> 00:25:30,000 It's not a crazy result, but it's super useful, 426 00:25:30,000 --> 00:25:32,580 because if we go back, let me see, 427 00:25:32,580 --> 00:25:35,100 do I have our old code here? 428 00:25:35,100 --> 00:25:37,470 So this is from the previous video. 429 00:25:37,470 --> 00:25:42,150 Really we could replace this entire block, 430 00:25:42,150 --> 00:25:44,700 we could draw less random numbers here. 431 00:25:44,700 --> 00:25:47,463 Now order doesn't matter 'cause we're fixing that. 432 00:25:48,660 --> 00:25:50,190 We've renormalize our rates, 433 00:25:50,190 --> 00:25:52,740 so we don't have to worry about all of this. 434 00:25:52,740 --> 00:25:56,853 So we have our clone factory and our different rates. 435 00:25:57,900 --> 00:26:01,323 Now, when we go over people, we're just gonna ask, 436 00:26:02,287 --> 00:26:04,320 "Is this person doing something," right? 437 00:26:04,320 --> 00:26:06,633 So I'm drawing one random number. 438 00:26:09,420 --> 00:26:11,260 This person is doing something 439 00:26:12,390 --> 00:26:15,030 at a rate c if it clones itself, 440 00:26:15,030 --> 00:26:17,940 l if it leaves, or c plus l delta t. 441 00:26:17,940 --> 00:26:19,803 That's my sum of Poisson process. 442 00:26:20,970 --> 00:26:23,220 Okay, I have this now. 443 00:26:23,220 --> 00:26:25,560 Now let's say they choose to do something, 444 00:26:25,560 --> 00:26:27,363 well, with probability, 445 00:26:30,840 --> 00:26:35,840 so with probability c over c plus l, 446 00:26:37,230 --> 00:26:38,283 it's a cloning event. 447 00:26:39,540 --> 00:26:44,163 So now, oh, this is a different sheet. 448 00:26:48,750 --> 00:26:52,110 So we say we add N of t as our variable. 449 00:26:52,110 --> 00:26:53,977 I got confused here. Oops. 450 00:26:53,977 --> 00:26:55,353 And ft. 451 00:26:56,250 --> 00:26:58,683 We go over people equal one. 452 00:27:00,660 --> 00:27:03,450 And this doesn't matter that much, but, okay, N of t. 453 00:27:03,450 --> 00:27:06,993 So it's a cloning event, so N of t goes to N of t plus one. 454 00:27:08,550 --> 00:27:12,240 And else, it's a leaving event, right? 455 00:27:12,240 --> 00:27:16,620 So N of t goes to N of t minus one. 456 00:27:16,620 --> 00:27:19,260 And now there's no more problem of which, 457 00:27:19,260 --> 00:27:21,570 are you cloning yourself at the same time as you're leaving? 458 00:27:21,570 --> 00:27:22,500 There's none of this now. 459 00:27:22,500 --> 00:27:25,200 We've sort of fixed this by using this property 460 00:27:25,200 --> 00:27:28,680 that a sum of Poisson processes is a Poisson process, 461 00:27:28,680 --> 00:27:31,170 and then dealing with it with our probabilities 462 00:27:31,170 --> 00:27:32,220 for each event. 463 00:27:32,220 --> 00:27:35,160 So that's it for this video. 464 00:27:35,160 --> 00:27:37,590 There's a lot more tricks that are gonna, 465 00:27:37,590 --> 00:27:39,450 or other problems that are gonna pop up. 466 00:27:39,450 --> 00:27:41,580 But these like are the key things 467 00:27:41,580 --> 00:27:45,630 that you have to think about, order of operation, 468 00:27:45,630 --> 00:27:48,960 of different mechanism and the scale of different rates. 469 00:27:48,960 --> 00:27:51,340 And the key mathematical tricks are 470 00:27:52,500 --> 00:27:54,750 this convergence of Poisson processes 471 00:27:54,750 --> 00:27:57,510 when you discretize with very small steps, 472 00:27:57,510 --> 00:28:00,390 and the fact that you can deal with competing processes 473 00:28:00,390 --> 00:28:02,993 as a sum of Poisson processes. 474 00:28:02,993 --> 00:28:05,443 So that really goes to the like mathematical core 475 00:28:06,479 --> 00:28:08,820 of these compartmental models, 476 00:28:08,820 --> 00:28:12,243 these simple flows of density and all that. 477 00:28:13,110 --> 00:28:14,280 So I really like this. 478 00:28:14,280 --> 00:28:16,470 If you want to talk more about how to implement them 479 00:28:16,470 --> 00:28:17,970 or how to do math around this, 480 00:28:17,970 --> 00:28:21,300 we can use some of our discussion time to go over it. 481 00:28:21,300 --> 00:28:23,500 But otherwise, I'll see you in the next one.