Transcription
Welcome back. So today, I'm going to tell you about one of my favorite probability distributions: the exponentially distributed random variable. Exponential distributions are really useful for modeling the time at which some rare event is likely to happen. So it's good for modeling the lifetimes of light bulbs, when a failure might happen, or when a radioactive element might decay, or when I might get, uh, seen by a human at the Department of Motor Vehicles.
For those of you who are not in the US, the DMV is the government organization that gives people driver's licenses, and they are notoriously hard, uh, to see a human in a short amount of time. Okay, so all of these, um, events—we're going to model the kind of time that, um, I might see these events occur—those are going to be modeled as exponentially distributed random variables. So I'm going to write down the definition, we're going to do some examples, and we're going to analyze and kind of understand what this means, okay, in words and in equations.
A random time, a random time T, is said to be exponentially distributed, or exponential, uh, with parameter Lambda. So there's a single parameter that determines this exponential distribution. I'll tell you what Lambda is in a minute, but it's a positive number. A random time T is exponential if the probability of this random variable Big T occurring at some little time t, some specific time little t, if this probability is, uh, Lambda e to the minus Lambda t for positive times, for T greater than or equal to zero, and probability zero for t, uh, less than less than zero.
Essentially, um, this probability density function for these, uh, times of events occurring is only defined for positive times. So we kind of assume that we start now at time zero, and the probability of something having had happened in the past is zero. I've assumed it hasn't happened yet, and this is the probability of it happening in the future. Okay, uh, and so let's think about some examples of this. I like the light bulb example. Um, a good light bulb should last for a really long time—thousands of hours, hopefully. Um, the light bulbs in my kitchen seem to fail the day that we put in new light bulbs. So, um, but you know, the studio lights here are high-quality lights that last for a really long time, and so, um, they're going to be—the time of expected failure is approximately an exponentially distributed, uh, random variable. And so I think I'll give you a couple of examples of what I mean by this, and I'll draw a picture.
So, um, maybe I'll just draw a picture of what this exponential distribution looks like. So here we have, uh, this kind of exponentially decaying, um, distribution here, where this is probability and this is, uh, little t time. And what this means essentially, approximately, is that for some little t, uh, time here, the chance of me failing in t between t and t plus, let's say delta T or dt, if you like, the probability of failing in this specific time in the future, that probability, uh, of failure that t is, you know, greater than little t and less than or equal to little t plus—I'm going to call it dt because we're going to do calculus in a minute—this is equal to, uh, Lambda e to the minus Lambda t * dt. And so the probability that this event happens between time a, uh, and b is going to equal the integral of this probability density function, Lambda e to the minus Lambda t dt, integrated between a and b. So this is how we work with these continuous, uh, probability distributions. We can compute the probability that this time occurs in a range of times as the integral of this, uh, PDF. And if you think about this specific probability, this is really for like an infinitesimal sliver of time, a little dt or delta t sliver of time, just like in calculus. You'll get used to this, and it might take a little bit. Okay, good.
So that's that's kind of how we define this, and, uh, and again, you can use this to model things like the probability that a light bulb will fail in 10 minutes from now, or 10 hours from now, or 100 hours from now. And, um, this is that useful distribution. Okay. Um, I'm going to actually write down a couple of exercises for you as the viewer to do, because you're going to get a lot more, uh, use out of this if you pause and do some exercises in the middle. So a couple of exercises—um, one of the exercises—exercise, not an example—is is derive the cumulative distribution function, derive the cumulative distribution function, which is, uh, this F of T equals, um, it's going to be the integral from negative infinity to T of your probability density function, P of T dt, and show—we're going to say dot dot dot—I'm actually going to tell you the answer: the answer is that this is is 1 - e to the minus Lambda t. Okay, so I want you to derive this expression for the cumulative distribution function from this probability density. That's one exercise. And a second exercise—this is one, the second exercise—is I want you to actually verify that this is a probability density. So verify that, uh, P of T is a probability density function; verify that the integral of this from negative infinity to positive infinity, if you integrate over all time, that this has probability that adds up to one. So I want you to verify that this is a probability that all adds up to one. So those are two really, really helpful exercises for you to get a handle on how to work with this exponential object. Um, good. And what else do I want to say? Um, I'm going to do a couple of examples here, but I think there's one other thing I want to show you, which is, um, and actually, I think I'm actually going to write out the answer here. So the integral of Lambda e to the minus Lambda t, this is, uh, minus e to the minus Lambda t, and we're integrating from a to b, so this would be evaluated, uh, from limits a to b, and so this is actually going to equal e to the minus a t minus e to the minus b t. That's a useful, uh, formula that you might want to compute the probability of your your time of event being between a and b. So that's if my event is between a and b, then this is the probability, the area under this curve, or the probability that my event happens in this range of time. Okay, so you can do all kinds of cool calculations with this; it's pretty simple, and we know how to integrate exponentials like that's the back of our hand. Okay.
Um, good. So why don't I do a couple of examples, and then I'm going to tell you some kind of strange properties of this exponential distribution. The most important property that we're going to talk about is this memoryless, memoryless property, that essentially, if I have already passed time s, if time s has already elapsed, then the probability of failure after time s is the same as this probability. So how do I want to say it? If my light bulb has an expected lifetime of a thousand hours, but a thousand hours has already passed, then the new expected lifetime from that starting point is still a thousand hours. So if a failure has not occurred, the probability of failure in the future is the same as it was at time zero. That's a wild property called the memoryless property of exponential distributions, and it's really profound; it especially is important in radioactive decay, um, but also all of these systems—that's an important property. Okay, so let's do, uh, an example here. So we're going to think about, um, the average lifetime of a light bulb. So let's say that the average, uh, lifetime of a a light bulb is, um, 100 hours. Okay, this is just a kind of crummy light bulb. So the average light bulb lifetime of a light bulb is 100 hours. What is the probability that a randomly chosen light bulb lasts at least 50 hours? Okay, so what is the probability that a random light, a random light bulb lasts at least 50 hours? Lasts at least 50 hours? This is a useful thing I might want to compute if I if I have an average, you know, failure rate; I expect my light bulb to fail in approximately 100 hours. How likely is it, or what's my probability that it lasts at least 50 hours? That's a very reasonable thing to ask. Okay. So, so I didn't tell you what this Lambda coefficient is, um, but this Lambda coefficient is is essentially, um, one divided by the average lifetime. Okay, so in this case, Lambda is going to be 1 over 100; it's 1 over 100, kind of meaning approximately I expect, you know, my rate of failure is 1 over 100. So in a 100, I expect a failure in 100 hours; I approximately expect a failure. Um, and in fact, the expected value—this is something you'll show—the expected value is 1 over Lambda, and also, um, I think the standard deviation is 1 over Lambda. That's that's kind of interesting too. So we have, uh, 1 over 100 for Lambda, and we're trying to compute now the probability that Big T is greater than or equal to 50. So we're trying to find the area under this curve to the right, uh, of t = 50 with Lambda equal, uh, 1/100, and this is 1 minus the probability that t is less than 50, of course. Okay, I could also compute the area to the left of 50 and subtract it from one; that's totally legit. Uh, and I'm going to use some of my basic formulas here, like my CDF formula, and I'm going to plug it in, and this equals, uh, e to the minus Lambda t, where T in this case is 50. So this equals e to the minus 1 over 100 * 50; that equals e to the minus 1/2. And you should compute e to the minus 1/2 or 1 over the square root of e and see—does that seem like a reasonable value? Um, I'm guessing so. e is about 2.7; its square root is probably about 1.4. I'm totally making this up; 1 divided by that is maybe 70%. This is just a total guess, but you know, a 70% chance that this lasts longer than 50 hours—that seems pretty reasonable to me if the expected failure is 100 hours. I'd expect, you know, a 70% chance that it lasts at least half of that amount of time. So this is a really easy way of computing things, and again, I want you to slow down, pause the video, work through the actual steps here. How did I—so this one's easy; the probability that it lasts greater than 50 is 1 minus the probability it fails less than 50, but this probability of T less than 50, that is the cumulative distribution function, 1 - e to the minus Lambda t, and I'm taking one minus it, so I get e to the minus Lambda t, where T again is this this little t is 50, so I plug it in there and I get e to the minus 1/2. Pretty easy to compute; with nothing special is happening, and it's really useful. This is—this seems like a very hard thing to compute: what, what's the probability that my light bulb lasts, you know, 50 hours? That seems pretty complicated, but it's a really, really easy, uh, formula here to work with. Good.
Um, let's do another example; I think it's important. Um, so another example: let's do a radioactive decay, because these are really, um, exactly exponentially distributed. So let's talk about the half-life of polonium, half-life of polonium, uh, polonium 210. Those of you who have watched my videos for a while know that I have a fascination with polonium 210 because this is often used—spies often use this to poison people because it has a really, really short half-life. So it has a half-life of 138 days. Okay, and so you can ask yourself questions, um, you know, what is the what is the, um, Lambda? What is Lambda if the half-life is 138 days? So the half-life specifically means the T where I have half of the probability to the left of T and half of the probability to the right of T; half of my expected mass of radioactive elements should have decayed, and half of it is still around. So the half-life is specifically the T so that this cumulative distribution function equals 1/2. So, um, f of T equals 1/2; that means, uh, f of 138 days equals 1/2, and so you can go through the math here, and you can compute Lambda in this case. I think we have, uh, 1, 1 - e to the minus 138 * Lambda equals 1/2, and so you can go through the math here, uh, subtract one from both sides, divide by negative, and you get, uh, Lambda equal the natural log of 2 / 138. So that's a kind of useful connection to something that you totally have intuition for is the half-life of a radioactive element. So this Lambda, this kind of rate at which I expect radioactive elements to decay, is related to the half-life in this way. Really, really cool. And you could ask yourself some real questions about this: What's the probability that one specific atom lasts one year, or 365 days? What's the probability if my half-life is 138; what's the probability that I make it all the way to 365? Um, how long would it take for 99% of a, you know, kilogram of polonium 210 to decay? Those are the kinds of questions you can ask. In fact, I'll probably have those as homework questions. Um, pretty, pretty useful stuff here. Okay.
Okay, so that's just about enough. I think I'm going to tell you the memory, memoryless property; I'm just going to write it down, and then I'm going to conclude. So the probability that t is is greater than little t plus little s, given that I know that little s time has already passed. So if I'm already if if if s amount of time has already passed, then the amount then the probability that I last another little t amount of time, this is just equal to the probability of Big T, uh, greater than little t. This is a very, very interesting notion. So if time little s has passed, if 100 hours has already passed, then the probability that I last another t time after that is essentially the same as the distribution if I restarted the clock right now. Okay, so that means that if my expected failure rate is a certain amount and I've lived a certain length of time, at that point my probability of failure in the future kind of always resets; it's continually resetting. So if my light bulb hasn't failed yet, then the probability of it failing in the future is still exponentially distributed, starting at time now. It's really pretty wild, and I want you to think about this, um, because it's both profound and very, very, uh, interesting and important. So if a certain amount of time has passed, then starting from now, the future probability of failure—my time always is continually resetting. Okay, good.
Um, that's it for now. I'll show you how this relates to the Poisson process, so, um, the exponential distribution is intimately related to the Poisson process, and I'll probably introduce something called the hazard function, uh, in the next lecture. Okay, thank you.