Hello. In the next few lectures, we're going to talk about probability. Remember, our overall goal is to create statistical models that emulate and learn from stochastic processes. So we need to start there. What is a stochastic process? We're about to define it and give an example so we can see what we mean. Once we've observed a stochastic process, we might develop some sort of belief about the behavior of that random process. From that, we need to turn our belief into a statement of probability. Once we have a probabilistic statement, we need to think about rules for manipulating it. And finally, we might need to think about how we can simplify it and we'll do something factory. Alright, let's get started, So first off what is a stochastic process? The term I've heard used is chance regularities. So what we're interested in observing are non-deterministic in nature. In other words, they exhibit some element of chance. We have no idea how to predict what the next outcome is if we're observing some sort of process with certainty. As an example, consider a game of dice where we are throwing to die and we're summing Mean the dots. Let's go ahead and simulate this in Python, I've given the code below as an example. Basically what we have our dice with 6 sides and we're going to choose randomly which side lands face up. We're going to split that into two die and we're just going to some some of the spots. If we go ahead and run this code and look at the first 10 rolls, we might get a table that looks like this. So, here's what the first die lands and here's what the second Thailand's and here's the some of the parent price okay? So the first roll lands as a 6 and 2, we sum it up, we get an eight. Okay, so we're sending here at the 8. And the question we might have is what's the next outcome? What's the next role going to be, where are we going to place our money? We have no idea, write each side. So each face of the die has an equal probability of landing face up. So we're stuck, right, we don't know how to predict the next one. Let's go ahead and roll the die again. We get a six and a two in this case and we get a 10, okay? From here, we have the same question. Well, what is the next role? So, you maybe this is the game in, and we're asked to bet on what the next Dyson that next role is. We would like to know, how to place our money in a way that, wins more than it loses, basically. And again, we still don't have any, reasonable way to, guess what the next role is. Similarly, sitting at the 10 we have no way to figure out what the last role, okay? So in this particular case, what we have is a game of chance, there's no, way for us to guess kind of what the next role is, at least right now. We observe all 10 rolls, we see them like so. In fact, we have 500 rolls of to die. And let's go ahead and look at a little bit more. Here's the first 50 rolls, I'm going to, I created a so called t plot where basically I plot the result from each successive roll like so. So the first roll was sum of eight and then 10 78877 and so forth. What we can see from this is that there's no clear pattern sitting anywhere in this plot. We can't guess what the last role was and we can't guess what the next role it will be with any certainty. Okay, so, we've got we've got an element of chance or randomness. In this game if we make a table of the outcomes of all 500 in this case throes of die, we might get something that looks like this. Okay, so the first column here is the possible outcomes. So in, die with phases one through six. Two of them, we can get the numbers two through 12 is the sum. Here's the number of times each sum occurred in this 500 throws have to die. We can see from this that two occurs left less often than, say, seven or eight, which occurs maybe at about the same frequency as say at 12. So there's some pattern here, that might help us with our, our betting. So if someone's going to say place your bet, you'd probably bet a two less often than you might bet, say, seven or eight. So there is some regular ness to this but it's still a game of chance. So let, let's change this plot into one where we Basically put it on its end and, plot the frequencies. We did that or this, simulated data set, it would look like this. So what we see is we have some sort of triangular shape to this graph. That really highlights the fact that there is difference in frequency of the various roles. That will show 6,7 and 8 as occurring more frequency frequently than, say the 2 and the 12. So this is chance regularities. The game is one of chance there's randomness involved, but there is a pattern that is emerging that we should be able to use. So we're developing a belief. We have an intuition on this game where we know that 6, 7 and 8 are going to come up more than 2 and 12. So how do we turn that into a probability statement? That's where we are going to go to next.