 So, now, we are going to move to further higher dimension. So, we started with single random variable, then we talked about vector random variables, then we are going to talk about about three dimension already covered right like vector means, up to m dimension is covered. Now, we will talk about infinite dimension. It is like I have collection of uncountably many random variables ok. And so, you will see that in most of the application that is what it is going to matter to you because you want to understand like how the suppose for example, you want to understand how the stock market is evolving right. You will be not interested in one day, you will be interested in how it performed in the last 5 years and how it is going to evolve in the near future. So, you have the collection of random variables which need not be some finite number there, it could be uncountably many. And if you are trying to like a find a trajectory of a particle or whatever at every point, every point of time you want to understand how that is behaving right and there are so many point such times. So, suppose let us say something is taking a trajectory and at every time you want to every time its behavior you are trying to control, but its behavior could be random at that point right depending on so many others. Now, you want to understand at every time instance you want to model it as a random variable. So, we will try to make this more precise. So, that will lead us to something called random processes. So, here earlier we had earlier also when we talked about random vectors we had collection of let us say finite number of random variables, but here that need not be the case. Here the indexing set t could be uncountable, countable, but it could be infinity. So, and if in this case if this t happens to be let us say integers, when I say integers let us say it is going to be like 0, 1, 2, 3, 4 all the way like that. Then we are going to say that then this random variable x is called discrete random variable and if this t is going to be let us say real line or some continuous interval then x is called continuous sorry discrete random process. So, it could be just like a time in the interval 0, 1 you take any point in the interval 0, 1 there is associated random variable. So, for example as I said if in the stock example case you said you took x t could be the value of the what is that our stock BSE stock index whatever let us say on every day. So, on every day you can count day 1, day 2 like this and its value x t is going to be random we cannot predict apriori what is that value. So, that will be denoted by this x t. If you give me day x t will tell x t is the random variable associated with that day. That is discrete random process. That is discrete random process, but suppose you want to understand let us say for example let us say you are moving in a vehicle and you are trying to accelerate your vehicle, but you are you are you are in a very uncertain environment right. At every point the velocity that your vehicle is attained could be random variable. So, in that case at every possible time in the time of interest you want to understand what is the velocity at that time. So, for example let us say I have interested in 1 hour of time. This could be all possible seconds from 0 to all possible minutes from 0 to 60 minutes and you can think of anything like either vehicle moving or I am hitting some object and I want to understand the temperature profile in that. At every moment you want to understand like what is the velocity or the temperature profile of that. So, in this case you can more that t here could be all possible values between 0 to 60 right. Every instant you are talking about. So, then in that case we are going to look it as a continuous random process. So, most of the things will be dealing with discrete random process, but you will also encounter many examples in real life where one has to worry about the continuous random process also. Now, we can interpret my random process in different possible ways. Suppose a random process I have like this. Now one possible way to interpret is as we already said we can say that for all t in t x t is a function from omega to r. This is just a random variable right. So, this is a selection of random variable. So, for every t this is like a random variable which is giving value to each of my elements in my sample space. Alternatively we can think it as this entire x itself is a function of. So, where it is going to give the value where for all t and omega in t, y x t of omega is the value of sample omega at time t at index t. So, you can think of this x to be now like this is random variable right. So, now it has you can think of it as two dimension to this one is the index and another is given that index what is the value it is going to take on a given sample. So, for x that is why we are saying it is index as well as what is the value it is going to take on a sample. If you tell x is collection of my is a random process corresponding to let us say behavior of my stock exchange or what is the index on a particular day. If I say on day 10th about this particular sample what is the outcome then this x is going to give this if you have to look at the on that on that particular day on the particular sample what is the value it took and this is how we can interpret this random process as well. Other way we can think about is so this is like fixing t you fix a time and then look at on that particular day how this value is going to change for each of the sample what are the possible values for each of the samples or you can fix omega a sample and then look at on this sample how it the value possible values on each of the days. So, for example let us say let us say you are interested in some 10 shares in the market whatever that companies are and you have a random variable which assigns for each of the shares whether it made a positive gain or a negative gain. Now, you can think of on a given day what is the outcome for each of my shares. So, shares is a collection of this sample space what is the value it took alternatively you can what you do you can focus on a particular share and then look at on different days what it took what is the value it took. Now you can think of X of t as a function in t and then to call X t omega as sample path corresponding to omega. For example, let us say I have this this is my time index and I am interested in knowing my X of omega of a particular sample. So, this sample may take value like this I do not know it may fall it may rise it may fall like this and this is going to be I am going to be yeah this I am going to just interpret as X of omega. So, as for different value of t it is going to give me what is the value taken on that sample omega here and this is we are going to call it as sample path. So, is this different interpretation clear? So, let us now rework them. If I am going to fix an omega and then I am going to see like as a function of t how it behaves. So, this is what I have done it for a continuous case if it is discrete I will have only certain points here because this t only takes some values right. Now you are going to look into like as this aspect what you do in this case if you want to draw this you are going to fix time and these are your omegas this could be let us say omega 1 omega 2 all the way up to like you had some n points and what is the value taken let us say this is some value here is some value and this is. So, this is like a single random variable right at a given time t and here it is given sample how it behaves at different points of time or a different index and here either you can now look it into in the joint space this and this given an omega. So, for h omega I can vary this t and get this graph. So, if you give me a t and an omega then I will come up with a particular point. So, suppose you give me some omega let us say you give me omega 2 and you also give me a t equals to 10 I will look at this graph for t equals to 10 and then I get the value right. So, and that second interpretation will just give you that fine. So, then we have so just an example suppose I have this w 1, w 2 are independent random variables such that let x n equals to summation k equals to starting from k equals to 1 2 let us say n omega k this is for all k. So, I am saying I am giving you collection of this random variable index random variable index at 1, 2 all the way up to infinity is this discrete or continuous random variable here. It is going to be discrete right because now I have index which are discrete points and now I am saying each of the random variables are independent and for them each one of them is such that it is only going to take two values 1 and minus 1 and probably what is it takes 1 is half and now further I am going to define another random variable x n where for all again let us say 1, 2 infinity. So, this is now some of these random variables is this x n now I have these two random variables one is this w collection of this w k's and another is this x which is collection of this x n are both of them are discrete random variables yes right because anyway w is only defined at 1, 2, 3 index like that and the x n is also this is x n is collection of sum of n w's and that is also defined for each of n integer valued. So, this both are discrete random variables now let us see how this look like W is a random process yes a random process right because each random variable I have defined like this and it is a collection of so many of such random variables. So, now let us understand how this w k of omega looks for some omega let us fix an omega and now let us look it as a function of k that is in time here. So, k is the index what I am doing is I am fixing a sample point and for this sample point it may happen that for the first one it could be taking 1 it could be then taking minus 1 then maybe taking minus 1 and then going plus 1 and maybe plus 1 again and like this plus 1 it could be this is one sample path. So, you remember I have defined a sample path here for this particular random variable I am now trying to draw a sample path maybe I do not need k here. So, this is like 1, 2, 3, 4, 5, 6 and I can go on each of this k my random variable is such that it is going to take either value 1 or minus 1. So, let us say when I perform my experience in the first round it took value 1 in the second round it took minus 1 and it again took minus 1 and it took 3 constitutive 1 after that and something happened subsequently I do not know what is that. So, this is going to give me a sample path of this process for a given omega. Now, let us try to draw my sample path for x now x is a random process which is a function of w. So, if I know this process should I also able to draw the sample for this ok maybe ok. So, let us see what will be the value of x of omega at any cost 1 it is going to be this and what will be 2 it is going to be 0 then minus 1 then 0 then going to be like this right. So, this is how we are going to get some picture of what is going to happen like on a given path. If you look at some fix a sample point and then we can visualize how on this point my graph is evolving as it takes different value. So, as I said for example, if you are going to focus on a particular share value you can now look at on each of the days whether it made a positive gains or negative gains and plot it like this and this could be like the cumulative effect. So, the cumulative effect is till date it made effectively positive gain or negative gain. So, for example, this curve here could be like on each of the days it is making positive gains or negative gains and here it could be till this point the cumulative gain is positive or negative ok. Then for such random process we are now going to define mu of x of t is going for all t is t. So, now I have so many random variables right one defined for each of the possible index now I am going to define mean for each possible random variable I have. So, that I have one random variable for each of the index right. So, for each of the index the mean value is simply going to be the mean value of that random variable. And this correlation we have now it depends on which two random variables we are talking about right. Now suppose if you are look you are talking about random variable at time at index s and t then you are going to denote that is now you have to specify which time which index we are talking about to calculate this correlation. So, if you tell me s and t are those indices then the correlation between that random variable is this. And similarly covariance of a random variables at indices s and t will be given by covariance of x s and x t. And then the CDF of this random variables and is going to be defined as now we have to when I am talking about this random process I have to tell which random variable I am going to talk about right and that is going to be specified by its index ok. So, suppose if I am looking at distribution of n random variables then I have to specify at what is the time what is the index you are looking at them. So, you are going to specify those indices. And then if in that case the CDF of this involving n random variables is going to be defined as the random variable at index x of t 1 taking value less than or equals to x 1 and the random variable at time x 2 taking value less than or x 2 like this all. Now to give a complete characterization of this random process you need to define this for all n n is what integer ok and for all x 1, x 2, x n belonging to R n right because we have now a collection of random variables to give a complete characterization of this random process you need to tell me if I am going to look at these set of random variables what is their distribution and I should be able to tell this distribution for any possible set of random variable you are going to ask me ok. That is why you tell me how many set of random variables you want to look at and you tell me which are the index what is the set of random variable to decide that you need to tell me the indices ok. And then ok we have to also tell this. So, then for all this and also t 1, t 2 all the way up to t n that is coming from your ok t to the power n because these are these many indices right. So, you have to tell me which are those indices you are looking at and what is the value you want to and I should be able to tell what is the probability that at that random variable taking value less than this particular number. So, I need to specify all of this to completely characterize my random process and this kind of things if you can define your this CDF for all possible value of n and for all possible indices and for it taking all possible vector like this if you this is called finite dimensional distribution. So, see like random process is a complicated thing right. There are so many random variables there and this could be potentially uncountable, but to define it completely you need to specify how any possible subset of this random variables in this random process are going to behave. If you are going to like if you if you cannot specify the way it behaves at some indices then you are not completely specifying me your random process. That is why to explicitly completely characterize your random process you need to define your CDF for all possible subsets for all value possible value it is going to take and also for all possible indices you have and that is so this FDT is what going to completely characterize your random process. And now this is the mean function here I am going to call mu of x t as a mean function for my random process x, why this is function? So, it is now it takes input t right it is going to change as your indices changes and this is so this is we are going to call a correlation function now. Earlier when we had a two any two random variables we know how to find the correlation right if you give me x and y expectation of x y is their correlation, but now I have I have so many of them not just x y I have x 1, x 2 all the way to infinity. So, you just tell me which two random variables you want to look at the correlation and I am going to compare. So, this is going to be a correlation function now and what is this it is now a covariance function. So, sometimes you may be interested in you may want to set s equals to t itself that is looking for a covariance of a random variable with itself. It is going to be variance, but in this parlance you can if you are looking at the same you can add auto covariance. So, at all right when I am going to look at the same time indices the time indices are not two different things same things. Yes, okay then we have this definition. So, if my random process is such that each of the component in this process has a finite second moment then I am going to call it as a second order random process, okay. So, this is just our definition. So, just one point I want to add here. So, this is I have this is a CDF right which I have defined for all possible subset if I know that my process is my random variables are all discrete, right. Maybe then I may be interested in only probability mass function equivalent of this. For example, if I have all my random variables are discrete then instead of looking for the CDF I may be just interested in x1 t1 all the way up to xn tn to be just equals to probability that x of t1 equals to x1 x of t2 equals to x2 all the way up to x of tn equals to xn. So, this is just like a probability mass function version if my random variables are all discrete. So, I am talking about two discrete things here. My random variable itself is taking discrete values and then the indices being discrete, right. So, my process is discrete random process if my time index is discrete, further if my each of my random variable such that it only takes value from a discrete values then I will be interested in only further this probability mass function in that case, okay, because this gives all the information. I do not need to go for this complicated CDF in that case, okay. And similarly we are also going to say that my set of any n set like this are going to be continuous here jointly continuous if they have a corresponding PDF the way we did earlier, right for a single random variable. If I can find a, if I am able to express this in terms of some function f of small f of xn in terms of integration where we did earlier then I am going to say this set of random variables are continuous, okay. So, we will just the n-th order PDF, okay this is just like completeness, okay. Now, I want to just let me complete this one more time then we will move to the next class. So, further we will study some more properties of this in the next class like stationarity and wide sense stationarity. So, before that I want to just tell you what I mean by a Gaussian random process, okay. I have talked about what is a Gaussian random vector, but now I have defined a Gaussian process then you may want to specify what we mean by a Gaussian process, right. A random process. So, now we are simply extended the definition of Gaussian, Gaussianity from random vector to random process by saying that we have a in the random process we have so many index random variable, but from this index random variable if any linear combination if the random variables this comprise of s are jointly Gaussian. That means if you take any linear combination of this random variable if they happens to be Gaussian then we are going to call this process as simply Gaussian random process and so we already know that for this in the random vector case, right. We denoted that Gaussian random vector as mu mu k and its pdf there dependent only on mu and covariance k, right. So, to define this Gaussian random vector I just needed to know the mean vector and the covariance vector. Now, what do you think I should know to define this Gaussian process? So, again maybe I just need to know what is the means for each t and maybe the covariance for each possible time index pair, right. So, for this, so the good thing about the Gaussian vector was it was parameterized but the parameters were just like the mean value and the covariance value. Now, to define a Gaussian random process completely so that I do not need to really look for all this finite dimensional distribution. So, maybe the parameters are just sufficient. So, what are those parameters I should be interested in to completely define a Gaussian random process. So, maybe one thing is you going to do this mu of x t for all t and then so maybe like we can just say that if you are going to like take any subset t1 of this time indices to define this joint distributions what all the things you need to know, you just need to know their mean vectors and their covariance matrix which is, so as I said to completely define a random process I need to define my finite dimensional distributions, right. So, for the Gaussian process what is this finite dimensional distributions? I know that if I take any n that will be jointly Gaussian, right for a Gaussian random process. To define that joint random process all I need is the mean vector and the corresponding covariance function and this covariance function can be expressed simply in terms of my correlation function and the mean value. So, as long as you give me the mean vectors as long as the correlation function I can I have the complete information about the finite dimensional distribution of my Gaussian random process because I know how to construct my probability density function for each of this n and I can do it for any possible n. So, is this fine? So, if you have a Gaussian random process its characterisation much simpler all I need to know is its mean vectors and its correlation function, but if it is not a Gaussian vector maybe things are bit more complicated I have to define finite dimensional distribution for all possible n. So, let us stop here, right.