Transcription
Welcome back. Okay, so we have introduced the concept of an exponential distribution, which tells you kind of the, the probability that some rare event, like a part failing, like a light bulb failing, will happen at time T. And we have also talked about Poisson processes, which are, you know, the expected number of a rare event in a given amount of time. These are clearly related, so your intuition should be, just like, you know, a light bulb should be going off, that these are related. And today I'm just going to show you how they're related—super simple—and give an example. Um, you could spend a lot of time, you know, really working through the details and the intuition of how these are related, but it's actually quite simple, so I'm just going to, I'm going to say it, and we're going to see it in an example. Okay.
So let's say we have a string of kind of random events that happen at some rate, like receiving emails. Okay, so we have a, we have a string of events, we have a string of events, uh, like emails coming in. Now we know that Poisson processes are for rare events, and at least for me, getting emails is not at all a rare event. I get like a thousand emails, uh, you know, overnight it seems, but the chance of getting an email in this infinitesimal sliver of time is still a rare event. So you can use Poisson processes for things like, um, arrival times, even if it happens pretty often, like me getting emails. So we have a string of events, like, uh, like getting emails, and I'm going to visualize this on a timeline. So we're going to have this big timeline, uh, from zero to T. So this is zero, and I'm going to visualize these, uh, these events as little X's. So let's say my first email, um, comes here, my second email here, my third email, fourth, fifth. I'm just making this up. Let's say I have some distribution of, you know, of actual emails arriving in time, uh, and let's for now say that this happens, um, at T1, T2, T3, uh, you know, dot dot dot up to TN. But, you know, let's just say that these are all of my times, and I'm going to define just a couple more numbers: the waiting time, um, between emails; we're going to call that waiting one, uh, waiting two, waiting three. So the waiting time between emails are these W, uh, variables here. Okay.
So the waiting times are clearly distributed as exponential, uh, random variables. So I'm just going to write, write this out: so the waiting times, the waiting times, uh, WJ are each independent, and they're distributed as exponential variables. They are independent. Now again, clearly that's a silly assumption for emails—like I get a ton of emails Monday morning at 9:00 a.m., I don't get as many emails at Sunday night—um, so this is kind of a fake example, but assume, you know, during the workday, maybe, maybe this is approximately true. So waiting times are, uh, independent, where they are distributed as exponential random variables, uh, with a parameter Lambda, where I'm going to define Lambda as essentially the rate of, the rate of the event. So Lambda is the rate of these events. So, for example, in emails, maybe Lambda equals, you know, five emails per minute. So they'll have units. Lambda might equal five if I get five emails on average per minute. So the waiting times are independent, uh, exponentials. This has been the topic of the last couple of lectures; we know about exponentials. What's interesting here is that the number of these events that will happen in a time interval T, that is a Poisson process. So let's say that I have some interval of time T. I'm going to make it little t; this is some, some interval of time, maybe this is 20 minutes. An interval of time, the number of arrivals, the number of, of emails or events, the number of events, uh, in time T, time little t, is Poisson with parameter Lambda t. So the rate, five per minute, and the amount of time T, this will give me the, uh, parameter that I'm looking for, is just Lambda times T.
Now this is kind of cool. So if I have these kind of randomly timed events that happen at some rate Lambda, then the waiting time between events is for sure exponentially distributed, but the number of events, that specific number of events I expect in an interval of time, that is Poisson distributed. And these are often called Poisson processes; these random time events are Poisson processes, um, because of this, this property here. Importantly, you'll notice that the waiting times, these are continuous random variables; the amount of time I wait for an event is a continuous variable; it could be 1.05 seconds in the future between my next email. But the Poisson distribution, this is a discrete random variable; it is an integer number of events I would be expecting to get, uh, in time T. Good. Let's do an example. Okay.
So let's actually do this exact example. So, um, example: emails come in at a rate of five per minute. Okay, so that means Lambda equals five. So what is the probability that you wait T minutes for your next email? Um, and I'm going to make that a blue marker. The probability that, uh, maybe I'll actually write it in words and then I'll write it in math. So the probability that, uh, I wait T minutes for next email, and clearly I'm, I'm talking about a continuous variable T, so I'm going to be talking about this exponential waiting time. This is the probability that, you know, Big T is greater than little t, um, and that is, you know, if you remember this kind of, uh, exponential distribution, this is one minus the probability that t is less than little t, and we use the cumulative, uh, distribution function. Um, essentially the answer is e to the -5T. That's the probability that I can wait T minutes. And notice that this gets smaller and smaller and smaller; the chance of me not getting an email for 20 minutes, that's a really big number, a huge number; it's not going to happen; I'm going to get an email in the next 20 minutes. Okay, good. Um, I can also compute—so this is the probability that I wait time t for my next email—what's the probability that I get no call, uh, sorry, no emails, no, no emails in the next, in the next T minutes? This actually seems like exactly the same thing: probability I wait T minutes for the next email, probability I get no emails in the next T minutes. But this is a discrete; I'm getting zero emails in an amount of time; that's my Poisson process. So this is literally the probability that k equals zero in t minutes; that's my Poisson process, um, you know, it's literally my function; I'm trying to remember what this is; it's like, um, Lambda to the k, e to the -Lambda T over k factorial is the, you know, Poisson process probability of getting exactly k events in this much time. So the probability of k equals zero, Lambda to the zero is one, zero factorial is one; this just equals e to the -5T. Notice these are identical because they're the same probability; the probability, the Poisson probability of no events in five, in, uh, in t minutes is the same as the waiting time being greater than T minutes; these should be the same probability. I can also calculate what's the probability that I get, uh, exactly one email in t minutes, um, and I think what that's going to be is Lambda to the one power, e to the -Lambda T over one factorial; that's going to be 5e to the -5T. So it's five times more likely that I get one email than no emails in the next T minutes. And where Lambda is the rate, um, five emails per minute on average. Really, really cool. This is one of my favorite things in probability is this connection between these two beautiful distributions, the Poisson distribution and the exponential distribution, and they're intimately, intimately related in this notion of a Poisson process of random events in time. Okay, thank you.