Transcription
Welcome back. So in the last lecture, I introduced this exponential distribution, which is a super useful distribution for, uh, computing the probability that some, uh, kind of rare event, like the failure of a light bulb, will happen at time T in the future. Uh, and we wrote down the probability density function; we kind of analyzed it a little bit; and we wrote down this property, the memoryless property of the exponential distribution, which is really, really profound. It essentially means that there is some probability of failure in the future, and it, um, you know, changes over time. And if I have already, if if s amount of time has already elapsed, then the probability of lasting another T amount of time is the same as the probability of lasting T starting at Time Zero. And that means that if I've already elapsed for Time s, if my light bulb has already lasted for 100 hours, then that after that 100 hours, kind of the clock resets, and my probability of it failing in the future, in in the future, is as if time was Zero from right now.
And this is a weird notion because we think, you know, sometimes we we count down these events, and we think if something has a one in, you know, if a failure rate of a light bulb is 100 hours, then after 100 hours passed, I'm like counting the seconds; I'm like, "Oh, it's going to fail any minute now." And that's not exactly true. The longer something lives, it doesn't mean that it has a shorter lifetime after that necessarily. So this memoryless property is really, really interesting. And one of the ways that that you can understand this, I think that, um, this is more intuitive for a lot of people, is in this notion of something called a hazard rate. Okay. So the hazard rate for an exponential distribution, uh, is the following. So we're going to compute the probability, uh, that we will last that we will fail in less than little t + DT. Okay, and I could basically just compute that it's the cumulative distribution function to the left of little t + DT, but I'm going to compute this given that I have already lasted Little T amount of time. So if I, given that my failure time of my light bulb is greater than Little T, meaning I've already lasted Little T amount of time, what's the probability of failing in the next DT moment of time? So this is kind of a a funny way of thinking about things, but this probability of failing in time t + delta T given that I have lasted T, um, it's a really interesting probability. Okay.
And so we can compute this thing. This is equal to 1 minus the probability, um, of T living longer than t + delta T given that t has already lasted, uh, Little T. This is again the probability of T being less than something is 1 minus the probability of T being greater than something; that's pretty okay. Uh, and this thing we can actually compute. So the probability of T being greater than t + delta T given that I've already lasted T, now I can use this memoryless property. So I'm going to switch to pink here, and if I've already lasted Little T, the chances of me lasting another DT is just, uh, 1 minus P that, uh, T is greater than a little delta T. So the probability that t is greater than delta T, and that is the cumulative distribution function, uh, evaluated at little DT. And the cumulative distribution function, I wrote this down in the last lecture, but I think it's like 1 - e<sup>-Lambda T</sup>; that's the the CDF, P, F, F of T, and that's the probability that t, um, this probability here. And so I can write this out, uh, and this equals 1 - e<sup>-Lambda DT</sup>. Okay, and we could stop here; that's totally fine. But there's a kind of, uh, manipulation you can do that makes this a little bit more useful. I can expand out this exponential in a Taylor series, and I can make a small DT approximation and get a really interesting result. So this is 1 minus the Taylor series approximation of e<sup>-Lambda T</sup>, which is 1 - Lambda DT + (1/2) Lambda<sup>2</sup> DT<sup>2</sup> - ... dot dot dot dot higher order terms in Lambda and DT, but we are assuming that DT is small; this is kind of an infinitesimal DT, and so all of these terms are really, really small, and these are the leading order terms; the ones cancel, and this is approximately equal to Lambda DT. So Lambda, this is known as the hazard rate, the the the the kind of risk, uh, associated with living another DT is this Lambda. This Lambda DT is the hazard rate of just continuing to exist. So this light bulb, in by the virtue of it not failing, it has this continuous kind of hazard rate of failure, Lambda DT, uh, and so in other words, the way we would write this is that the probability of, um, T being between, uh, Little T and little t + DT is equal to Lambda times the probability of Big T greater than T DT.
Okay. So this is kind of an outfall, and you probably want to pause and think about why this is true, um, kind of based on this reasoning here, but this is a really, really useful property. You use the memoryless property to compute, given that I've already lasted T, I've already lived T, what's the chance of me living another, uh, of of me dying in the next delta T? This is kind of the chance of me dying in the next delta T is Lambda DT; really, really interesting property. You could write down a differential equation and actually relate these if you wanted to, and you could think about it in context of radioactive elements; that would be a pretty cool, pretty big homework problem that would be like that would take a little while to really wrap your head around, you know, this hazard rate being Lambda DT and radioactive elements and deriving this, uh, probability density function. But if you do that, you'll have, you know, a super, super deep understanding of this exponential distribution. Okay, thank you.