Theory of Stochastic Processes belongs to the field of mathematics known as Stochastics. Basically, there are three disciplines which are taught in almost all universities and which are related to Stochastics. The first one is the probability theory, the second is mathematical statistics, and the third one is the theory of the stochastic processes. And if you are familiar with all of these disciplines you can take much more difficult parts of Stochastics like jump processes, stochastic calculus, stochastic differential equations, also population dynamics, jump processes and many others. Well, we will somehow discuss many more difficult ideas within this course, but now let me explain what is the difference between these three parts of Stochastic world. Well, let me start with probability theory. Assume that we have a pond with fishes, and when we speak about probability theory, the most typical task in the context of this example will be to analyze the amount of fishes in this pond at some given time moment. And the most typical situation is when you consider distribution of this amount, and if you know the distribution, you can further analyze this amount, for instance, calculate mathematical expectation, variance or even to find a limit law for the variable capital M. Now, let me move to the mathematical statistics. Mathematical statistics solves problems which are upper side to the probability theory. Namely, one can ask how to estimate the amount of fishes n by providing some statistical experiments. I would like to show one approach which can be used to solve this rather practical problem. For instance, we can catch some amount of fishes. Say capital M fishes and mark them, and then we put this amount of fishes back in the pond, so the moment we have capital M, marked fishes and capital N minus capital M, unlabeled fishes. Next, we catch some amount of fishes once more. Let me say small n fishes. From this amount, there are some marks and some unlabeled fishes. Let me assume that we have a small m, marked, and n minus m, unlabeled fishes. The question which we should take into account is what is the probability that the amount of marked fishes is exactly small n. This can be calculated using some very simple considerations from the probability theory. Namely, this probability is equal to C capital M small m multiplied by C capital N minus capital M, n minus m and divided by C capital N capital M. And afterwards, we can repeat this experiment many times, namely we get the values m1, m2 and so on, mq, and for any m, we can calculate the [inaudible] probabilities. And afterwards, we can consider as a log-likelihood function which is equal to the sum K from 1 to q, are probability that the amount of marked fishes is equal to mK. And here, if you look attentively at this formula, you will get the following conclusion. That almost all elements of this formula are exactly known. For instance, you know capital M because you know how many fishes you'll catch at the first time. Then you know also small m and small n because these are the results of your experiment, and the only value here which is unknown is exactly capital N. And therefore, you can deal with this parameter as an unknown parameter and you can just maximize log-Iikelihood malfunction with respect to capital M. This means that you will get an estimator of capital M and this estimator is known as maximum likelihood estimator. Okay, this was a practical solution for a very simple task, and in this context, we can also introduce the notion of stochastic process because this m amount of fishes in the pond, definitely depends on T events on time, and therefore, we can ask the same questions, but taking into account that the amount of fishes changes and this is exactly the objective of the theory of stochastic processes.