 Hi everyone, it's MJ and in this video I want to talk about Komogorov's backwards differential equation. So earlier we've spoken about his forward differential equation, now we're looking at the backwards differential equation. And it's given by the statement here saying the derivative of the transitional matrix is equal to the generator matrix times the transitional matrix. So if you want to know what those matrices are, we spoke about it in the previous video, this is your transitional matrix. This is your generator matrix and the big idea was that your transitional rates are your derivatives of your transitional probabilities. So using that theory that we spoken about earlier, let's look at the Komogorov's backward differential equation. And another way of looking, like we said this is the transitional matrix, this P of t, you can think of it as the matrix of all the various probabilities. So what we can do is we can almost write out each of those probabilities as the following equation. So we can say the derivative of each of the probabilities is going to be the sum of k element of s of the force going from state i to k times the probability of going from state k to j over time t for all states i, j elements of s. Now, this is very abstract. It's like, okay, well, what's what's going on. So what we're going to be doing is let's look at some examples and that should help you understand this Komogorov backwards differential equation. So first thing I want to do is let's look at going from healthy to having the corona virus. So let's maybe just bring up what our transitional diagram is as well as our generator matrix. Now, what we're going to be doing in discussing this is using this Komogorov's backwards differential equation to kind of explain or figure out what these probabilities are. So what we're saying here is the derivative of the probability of going from healthy to corona over time t is equal to negative the sum of getting the corona virus and dying times the probability of going from healthy to corona over time t plus the rate of becoming infected times the probability of staying with the corona virus over time t. So let's maybe look at that again. The transitional rate of getting corona over time t. Okay, so what is the probability that we're going to get the corona virus over time t given that we're healthy? Well, it's going to be the same as staying healthy and then getting the disease. So the rate of staying healthy times the probability of getting corona over times t plus the rate of getting corona times the probability of staying sick over time t. Let's look at another example where we go from healthy to dead and the idea here is that we can go from healthy to dead three ways. Okay, we can stay healthy and then die. We can die and then stay dead and we can get the corona virus and then die. So there are three things that we need to take care of and let's maybe talk about that. So the transitional rate of dying over time t is equal to the rate of staying healthy times the probability of dying over time t when healthy plus the rate of getting corona times the probability of dying over time t when infected plus the rate of dying when healthy and then staying dead, which has got a probability of one. Another one that we can look at is what happens when we are staying healthy. So if we're staying healthy, then we have, well, first of all, we donate it by drawing a line over H H and what that line over it means we're always. So it's the rate of always remaining healthy over times t is equal to the rate of staying healthy times the probability of remaining healthy over time t. So if we come back to our graph, we stay healthy and that's it. We're not moving to corona and we're not dying. So that's why it's a lot simpler. It's the rate of staying healthy times the probability of remaining healthy over time t. Now, this probability of remaining in the healthy state over time t is also known as our survival probability. Remember, this is something that we spoke about way in the beginning of T P X. So it is a survival probability. Now, one of the things that we can do about survival probabilities is we can say that they're equal to the following. This is using the common root of differential equations. They might be saying, well, what does that lambda I stand for? Essentially, lambda I is going to be equal to the sum of all the other transition rates or it's equal to the negative transitional rate of staying in the state. So think of lambda I as the total force of transitions out of state I. Now this can be useful because we can use it for various things. For instance, because if we had to, let's say, look at this over here, that should remind you a little bit of the exponential distribution. We can see the distribution of remaining in state I is exponential with lambda equal to lambda I. And because it's exponential, we know quite a few things about it. For instance, we can now calculate the expected remaining time in state I is going to be equal to 1 divided by lambda I. Now the probability of, let's say, moving to corona or when leaving the healthy state is given by the following formula. It's the rate of getting corona virus divided by the force of mortality plus the rate of getting the corona virus. And we can then also estimate these rates using the following formula. It'll be the random number of transitions from healthy to corona divided by the random waiting time in state I. And essentially this is Markov jump processes at a very much an introduction. Like I say, I do expect you guys to now go through the notes, read it for yourself and you will see that. Okay, this is what all of the stuff means. I've got a bit of an introduction. Let me now go read through the notes and fill up any other blanks that come in and attempt some past paper exam questions. And that very much ends off the Markov Jump Process video. Thanks for watching. Cheers.