Hello and welcome. Thank you so much that you are still with us. This week, we will speak about Poisson processes and some related models. But before we proceed to the topic of this lecture, I would like to show through the course the main aspects of the renewal theory, which was started before. Actually, a renewal process is a discrete time process, which starts from zero at a time point M. It is equal to it's value in the previous time point M minus one plus Xi n. Where Xi 1, Xi 2 and so on is a sequence of independent identical distributed random variables with positive distribution. With any arrangeable process, we're going to associate the so-called counting process N t, which is equal to the maximal index of S such that S k is less or equal than T. Renewal processes, as well as counting processes, have a very clear interpretation coming from marketing. Let me show how it works. I will plot a typical trajectory of the process N t, and explain this trajectory of the point of view of marketing. Imagine that you have a shop where we sell some goods. And at the time perform zero, we don't have any good purchases in our shop. We start our sales actually. Let me denote the axis by t and N t, and we wait some time, which is modelled by Xi 1, until the first customer is coming. So this Xi 1 is time from the beginning of our sales till the arrival of the first customer. Process N t is equal to zero between zero and the arrival of first customer which is note by S_1. Afterwards, the first customer is arriving and our process N T is equal to one. So, we should wait some time, which is equal to Xi 2, until the second customer is arriving. From S_1 to S_2, the process N t is equal to one. Afterwards, we shall see some more time, Xi 3 till the time moment S_3. And between this S_2 and S_3, our process is equal to 2. So, it is equal to 1 at a point S_1. It is equal to 2 in the point S_2. And it's more than two, it's equal to 3 after a 3. So N t is the amount of customers at time moment t. A question which is interesting both from theoretical and practical points of view, is how to calculate the mathematical expectation of the process N t. The context of marketing, this expectation is actually means the amount of customers at time moment t in average. For a theoretical point of view, this mathematical expectation is equal to this infinite sum. Here, F stands for the distribution function of the random variable Xi I. And N star means N times convolution of the function F, in the sense of distribution functions. This formula is very nice, it's very beautiful, but actually it isn't easy to apply this formula in practice. Because even if you know the function F in closed form, it's very difficult to calculate its S convolution in general form. And moreover, it's very difficult to find a limit of this infinite series. A moment method which I presented to you on our previous lecture, is based as the so called Laplace transforms of the function U. Let me recall that Laplace transform of the function U, is the integral of R plus exponent in the power minus s x U of x d x. And this method, uses one more assumption on our process namely, we assume that Xi I has an absolute continuous distribution, that is possesses some density which I denote by P. So P is derivative of F. And it turns out that the Laplace transforms of the functions U and P are related to each other via the following formula. And this formula gives rise for the following method. So, if you have probabilities distribution function P, then we can calculate the Laplace transform of P. Then we can use this boxed formula and recover the Laplace transform of U. And afterwards, we can use the inverse Laplace transform and get the mathematical expectation of N t at this function U. It was well with any probability of densities speed. But there is one difficult place in the scheme. The scheme, this difficult place is hidden in the last arrow. Actually, the inverse Laplace transform was a rather difficult operation. And normally, you should just guess which function U has this Laplace transform. On our lecture, on today's lecture, I will present you one particular case of the Poisson process for which the functions, the distribution functions of S n and N t are known explicitly. And in this case, the theory of finding the mathematical expectation of N t is not new at all. Let me now give the definition of the Poisson process.