 As of now, the way we define to this random process, they are very general notion, very abstract general, it just said it is going to capture something, but often to model something and to analyze this, we want this random process which have bit more structure in them. So for example, when we said random variable and distribution, but any random variable if I just take any arbitrary distributions, maybe it is hard for me, then that is why we define some special distributions, we said uniform distribution, then said Gaussian geometric, different distributions which are helpful to model the reality to somewhat extent, but they are also bit more structured in the sense we can analyze them system when we model using these distributions. Now similarly, when we are talking about different random process, any arbitrary random process if I take, maybe I may not able to analyze my system at all. So I would like to now look for certain random process which are more structure in it, but still capture my reality, then those using those random process maybe I can try to understand my system and get some useful insights. Now we are going to now start focusing on special type of random processes. So we will see like today and tomorrow we are going to talk about one special random process called Poisson process and later in the second of this course, we are going to focus significantly on a particular random process called Markov processes. Now let us say Poisson process. So to define this notion of Poisson process, we have to understand two properties called notion of independent increments and another called accounting process. So let me first talk about independent increments. I mean most of these things like you will find or start relating these things actually when you get into modeling business and try to analyze. If you just get into just some coding business, you will never appreciate any of these things. If you want to real system, you want to analyze, yes you need all of these things like you need systematic tools to understand any system and you have some common language of talking to other guys who are also doing or understand what you are talking about. So you need all these things. If you are not going to do any system analysis or modeling, maybe in your life you will never appreciate this or you will never find the use of these things. So what I mean by independent increments? Suppose let us say I have a process xt and I look at over an interval let us say ab. So let us time being let us assume that this is a continuous process. So my t is continuous here and I am going to take some interval. Then I am going to look at this difference that is the difference in my random variables at this on this interval. So I am going to look at my random variable at this time index b and at the beginning of this interval at a and then look at what happened in this. So suppose your process is like something like whatever it is and I took this interval and I want to see that between this interval what happened actually I increased or decreased whatever you want to understand. So let us say you are interested in this interval and now we are going to say that random process okay. So what we will do is if we have a random process let us say and you take any n and you take n indices such that this indices are arranged in increasing order. The first index is t0, second is t1, t2 to tn and now if you look at this differences between this interval the process is x t1 minus x t0 then x t2 minus x t1 like that. Now if these are mutually independent okay what I mean by mutually independent you understand the meaning of mutually independent if I take this it is independent of all others. And similarly for every point if they are mutually independent then I say that it is independent increments okay. For example here if I would let us say this is my, I will take another interval another interval here I mean. So this is let us say t0 and this is t1, t2, t3 and let us say this is tn. So what if I take the increment in this and look at the increment in this interval like each of this interval if that is going to be independent then I am going to call this as independent random process with independent increments. So notice that these intervals need not be of the same length one could be larger here one could be smaller. So that is why like this is any set of indices just that they are in increasing order and if you are going to look at the difference they should be mutually independent okay. So now something called we looked at something called counting process. So counting process as the name indicates it does the job of counting. So for example in this graph I have shown here it could be like a plot of something like the number of people that have entered at this time and let us say nobody will use once they enter luckily let us. So this is going to be increasing curve and this maybe let us say if it is an increasing curve so they may be like you this if I am going to look at this point or this point you can think of like how many people by that time have entered are already there and that could be like thought as a counting. So that we will try to make it some more formal so this is just like a general graph. So let us say we are going to say that function we actually already said this. So you take a function f which will give you at any time any positive real number integer valued outcome and it is such that it is non decreasing that it keeps increasing and it is right continuous. We already understand what we mean by right continuous right. Such a function is called as counting function okay for example of a counting function could be let us say I am counting number of students that are arriving into the classroom okay let us say all my arrival happens between 5.30pm to let us say max 5.35 or 5.40pm and let us say you guys enter one at a time okay and then at each of this time second maybe I will have jump like this it remains like this maybe like this here that nobody comes becomes like this then some people start coming that that that okay. So such a process such a function can be a counting process right by our definition it is non decreasing and I can make it right continuous always by defining the point to be at this to be at this point and I can also make this to be a 0 at f of equals to 0 okay fine. So now such a function counting function can be represented in many, many ways one possibility is suppose I am going to denote this okay this is t0 t1 t2 this is t3 and this is like t4. So t1 is the time instant when the first entry happened or when the first count happened. So t2 in the time when the second entry happened so I can on my x axis I could denote my time like this right so that whenever entry happened that time index I will look into then I will call appropriately if it is the fourth entry then I will just denote it as t4. So then my f function can be represented in terms of this ti's right so I am saying thing right like if you tell me this exact points t1 t2 on this x axis I know that on this points a jump is happening and then I can construct this function from that so for example like if you just tell me anything and just tell me okay here here here here you just give me this t1 t2 t3 t4 then I know my function look like here here here here here here here here so I could reconstruct my function. Another possibility to represent the same function is instead of giving this time at which it is arriving just give me the entire counting time for example okay after how many time how much time from the origin the first arrival first count happened let us call this un and since the first count how much time it elapsed before the second count happened so that could be u2 and after the second count how much time it elapsed before the third could happen this is u3 like this right so you could also express if you I could also write this in terms of ui's where ui's are this inter count times okay so how is this ui's related to ti's so what is u2 here t2 minus t1 right and by default t of 0 is going to be taken as 0 so then I said u1 here this is going to be t1 minus t0 right t0 is always going to be origin for us okay so the same f can be expressed either in terms of this count times or in terms of this inter count times right now so how to make this explicit for example if I suppose if I am given given let us say I have been given this tn's count times how can I construct my f function from this so I want to construct f means so suppose let us say you have been given this time slots and I will ask you to tell me what happens to this function at time t so give me the value of this f of t so that means you have to basically count how many counts have happened before that and that is going to give you the value of function f of t right so the formal way of writing that is indicator that t is greater than n equals to 1 to infinity so in this case let us say your t is here we know that this t is going to be greater than t1 that will add one here it will also going to be greater than t2 this will add another one here this is going to be greater than t3 we will add three here but t is not greater than t4 I am saying they are all going to add 0 0 so here you will recover that the value is going to be 3 in that case okay now if I have only given my ft function like this how to recover this count times so I want to now find out tn from my ft function so how you are going to represent this this is going to be minimum value of t such that f of t is going to be greater than or equals to n is this correct try to digest this so let us say I have been given this function ft now from this I want to understand when the nth count happened right so just look when this guy f of t is going to take value larger than n but look at the smallest value of that t so that is going to give me my tn and I have already represent represented ui in terms of this count times but I could also write tn in terms of my inter count times how can I write tn in terms of uis so it will like suppose if I want to get t4 all I need to do is this this this this uis I need to add this is ui i equals to 1 to n right so now coming back to counting process so all of you understand what I mean by a counting function right now we have already defined what is the counting function it has to satisfy like non decreasing property and write continuity property and I already defined you what it mean by sample path of a random process the sample path of a random process is if you fix an omega and look it as a function in t that graph is we called it as sample path right now what we are saying is if the random process is such that if it sample path is a counting function then we are going to call it as a counting process is this clear so each of the sample point has to be a counting function then we are going to call that process as a counting process that is why like you can when you see example it will be more clear this is just a definition so then we have defined two properties one is the independent increment property and another is a counting property so there is one distribution what I already called as Poisson process which is based on these properties and it comes very useful when we want to model many many things so let me define that process now which is based on these two properties okay just what random process means is for a given lambda it is a counting process that means all of its sample path has to be a counting function then it has independent increments so if you look into different number of counts between different disjoint intervals they have to be independent and third if you look at this random variable first take some t and s and look at the difference so the way I defined is okay it should be empty minus ns that is the random variable at nt and ns they have Poisson distributed we all know what is Poisson distribution right but with what rate that rate depends on the length of the interval multiplied by lambda so if these three properties holds we are going to call this process as a Poisson process and this Poisson process have very interesting properties which we will talk in the next class.