[MUSIC] Hello and welcome to the course of stochastic processes. The theory stochastic processes is a field of mathematics which deals with probabilistic systems evolving over time. And first of all I would like to explain what does probabilistic system mean? The simplest probabilistic system appears when you toss a coin. Well, tossing coins is widely used in various game to determine the first player. And as a result experiment you get eyes at ahead or a tail, of course you can see it's just a mechanical action. Simply accelerate the coin and it move according to some physical laws, at least according to gravity and air resistance. Nevertheless, the complete description of the trajectory is rather challenging. It is much more pleasant to forget about physical background and just to consider the results of this experiment, it can be either a head or a tail. And it's very common to assume that both outcomes can occur when it's probability one-half. In some sense, you move from deterministic world to the stochastic world. And I would like to explain what is the difference between these two worlds. Let me draw one simple table. If here I have the deterministic world, And here, stochastic world. So let me start with single variables. For instance, one can analyze the course of a disease, and as a single variable, you can consider just a temperature of a sick man in the first day of illness. Now the deterministic world, this is just a real number. For instance, you can measure a temperature of a given individual and get the temperature in the first day is 39 degrees. But now, let me think about all sick men with this disease. I cannot manage the temperature for all of them. And therefore, I should use some probabilistic structure. It's very common to assume that there is some mean temperature in the first day of disease, and also some variances temperature. In this case we'll have a random variable, With some distribution. And according to this distribution, we can calculate mathematical expectation, variance, and all other characteristics of this random variable. Secondly, let me now consider dynamic variables, or in other words, variables changing over time. For instance, if you continue this example, you can ask what is a temperature in the first three days of disease. If we are in the deterministic world we can just once more measure the temperature every day or every hour as you wish. So we get actually a function from R+ to R. This function can be equal to 39 the first day, to 28.5 the second day and so on. As for the stochastic world, we should first of all take and pick out that at any day the temperature is a random variable. And secondly, we should definitelynot forget about the situations at this temperature is evolving over time. And so for of the situation, was just a stochastic process. So a stochastic process plays a role of functions, but let me say random functions. Okay, we'll return to this table a bit later, but now let me explain how R curves helps to understand the origins of the stochastic world. [MUSIC]