 So, we have today we are going to start with simple stuff, some simple notation, some basic examples. So, we will I will for the most of the initial lectures that I am going to give before the midterm, I am going to largely follow it from the book. The first book I gave as a reference last time that is introduction to probability models and by random process for engineers, okay fine. So, as this we say this course is about introduction to stochastic models, the book the first reference I said that says introduction to probability models right. So, what is the difference probability and stochastic they are the same. So, what we are interested here is, so okay the book says introduction to probability theory and our course is titled as introduction to stochastic models. Theory is used to analyze models right develop models. So, what is this probability theory we are going to develop to study models right. Many things we are going to face in real life they need not be certain right there is always randomness associated with them. So, is there a way like I systematically model them first thing is I need to model them if I can systematically model and then I can analyze and then whatever the analysis I have done I can tell that analysis to somebody else and that guy will understand it if you can you can talk to him what is the model that you took it. So, fine there will be like if you give me a realistic scenario like nobody is God right like you cannot perfectly model it. So, you are going to model it according to some set rules which we are going to study in this rule. So, if you are going to do that then it is possible for you to explain the model you have developed and what all the analysis you did and then that guy may not or may agree with your analysis depending on whether he agrees with your model or not. So, broadly we start analyzing or we start to study some basic terminology that will be useful for us to start talking a stochastic model. That means, yes I am trying to model a system there is uncertainty but I want to still model it and what is that notion of this stochasticity we are going to make precise by going through sequence of terminologies now. So, the first thing we are going to study is, so I am going to call anything so even you are going to model something yes there are set of possible outcomes possible with that I am going to denote the collection of set of all those possible outcomes as sample space and I am going to denote it by S. So, whenever now I call experiment it is implied that I mean an random experiment that is the outcome itself you cannot upper you know what is that it is going to take. So, ok let us now try to distinguish what is a random experiment and what is a deterministic experiment can you tell me an a certain thing in life which you are sure shot to know what is the outcome is going to be determining shape of earth ok. Let us say I mean God has built something I mean that nobody can change these are things certain things are fixed right. So, for example loss of physics they are fixed right like you know that is how it has to behave these are fundamental laws you cannot change. But now let us say you have a toss of a coin can I consider this as a random experiment. So, the outcome could be head or tail right. And each of head and tail can happen I mean I do not know like when the head I cannot operate when you toss a coin I do not know whether it is already going to be head or coming to or head or tail and that is why. So, fine that is possibly the reason where is the coin toss used to decide who is going to bat first or which quote you are going to choose or who is going to serve first because that is possibly a very random variant and maybe it is not going to bias towards anybody. We will talk about this what I mean by bias, but at least a priori we are not going to be saying ok this guy is going to be favored this guy is going to get whatever he prefers. So, we are going to do that randomly. So, we are going to denote sample space by omega and depending on what is that random experiment we are talking about this could be a different set. For example, if I am going to talk about coin toss what is this set is going to be two outcomes right just it is going to be head or tail right now let not put them in flower bracket just let me write this head or tail. Other you might have already talked about it 1000 times like toys toss of a dice right when you are going to throw a dice what are the possible outcomes 1 to 6 that is going to be our sample space. So, this is I am this is a random experiment because if I throw it a priori maybe I do not know what is the outcome that I am going to see out of this. So, let us say that is. So, when I asked you can you give me an experiments where the outcome is almost deterministic it is deterministic you have to think right like it something immediately does not come to my if I ask you ok give me a random experiments I mean everything that comes to mind is possibly random right possibly like there will be a class today it is random I do not know what happens if that lecture gets out he may not come for class or you you may for some reason may not attend the class or whatever various reason that is not under your control. So, many things are random in life and if you want to model them or analyze them. So, we need to have a systematic you have to develop it systematically. So, one concept we are going to use this notion of sample space. So, instead of one coin suppose I am going to take two coins and throw them I mean in each trial I am going to toss two coins what could be possible outcomes for me. So, I can see possibly head head on both set on both coins I am going to see tail tail or past his head other tail or and similarly if I am going to throw through dice what is the possible outcome 36 right may be the better way to represent that is the matrix where it is. So, these are very toy trivial examples where as soon as I tell what is the experiment you right away know what is the outcome is going to look like ok. So, we will leave up to our convenience how we are going to represent it when it is as simple as I did not tell you write it like this when it is more convenient for us to write it in matrix form we will write it it is just like collect. So, these each one is an outcome here. So, we are going to say each element in this omega is an outcome we are going to say. So, this said are simple things it is pretty easy to write down, but often I mean you are not going to model what coin toss right like you are learning this goes to do much much more complicated things. For example, you want to model how the weather prediction you want to model my trajectory of a missile or like my trajectory of a satellite whatever the things or whatever like if I send a signal whether my signal will reach my destination or if I am going to take a particular route whether I am going to reach my destination within a stipulated time many things you want to model. So, in that case we always not worry what is the we are not always in sit down and first write what is the set of possible outcomes in this. So, for example, if I am interested in let us say find average height of all Indians. So, then what could be out sample space in this? Yes it is random I am asking about what are the possible outcomes yeah any value right like maybe the shortest person is let us say from 2 feet I do not know if it is or and I am saying about others let us say some of average age of Indians who are 21 and maybe like something like 7 or 8 feet all numbers in these things are possible. So, you are not going to just like maybe write along exhaustive individual value, but maybe write a range or let us say you are trying to model weather forecasting there I mean millions of millions of parameters will be involved in how the things evolve and based on what is the value of these millions of millions variable different possible outcomes are there. We just do not try to enumerate them, but it will be on our back of our mind like what is the possible outcome is ok. So, whenever it is possible we just enumerate them, but whenever say sometimes I may just say this is sample space, but that sample space may be like abstract we may not we may have its visualization, but we may not write it formally on a paper or on a board ok. Now comes event. So, once we are given a sample space then we talk about events, what is the event? For example, in your coin tossing problem what are the possible events? The possible event in a single toss of a coin either it is head or tail and when it is a cost of two coins the event could be like both showing up head, both showing up tails or one of them showing up head tail and otherwise ok. And we are just simply going to denote an event by like this and talk to you and then you will be always interested in the questions like what is if I want to do a two coin toss what is the likelihood or probability that you are going to see some value that the outcome is going to the event is going to take one of this possible values. So, let us say E and F are two events then we say E and F are actually disjoint what is this? It is a null set, what does null set means? It does not have any element ok. So, we are saying what does the interpretation of E intersection F equals to null set means? So, they do not have any common element. So, when they do not have any common element we exactly call these two events are mutually disjoint. For sets we call it mutually disjoint, but these are events we are going to call them as mutually exclusive. So, similarly let us say we are going to have E 1, E 2 like E n like let us say these are collection of events they are all coming from a same sample space. We can talk about their unions, we can talk about their intersections and we can take about any combinatorial structure that is the union of some along with intersection with others and all. For example, if you want to let us say I want to take union. So, what is this? So, I am just taking union of all the sets. Now, suppose let us say E 1 is H H and E 2 is HT. What is the union of E 1 and E 2? So, union of H 1 and H 2 says what? The first outcome is head that is right like you do not care about second. Second can be either head or tail, but you want the first outcome to be head. So, in that case these E 1, E 2 could be some events in your space, but you can combine them to derive more events and similarly you can look at intersections. So, I could write it as intersection. So, what does intersection of set of events give you the common element? So, in the same example, this H H and HT the intersection will be what? So, notice that when I said E is a subset of omega. One element can also be a subset. So, here these are the individual elements in the set omega. For example, E 1 could be this element and this element and E 1 could be or H 2 could be these three elements. So, now when you are going to take intersection of let us say just two element in this there is nothing common between them. But suppose you are going to define E 1 to be this set and E 2 to be this set you can do that like you can take this because both are subsets of my omega. What is that intersection is going to give you? It requires that both tosses give me tails. Now we will get to the notion of probability and probability of events. So, we will make the notion of probably precise more, but let us try to write down informally what we mean by probability of events. So, you take an so hence more when I say an experiment or random experiment that will always come with an associated sample space. So, when I say random it is going to take all possible some possible outcomes. I am going to call that as sample space and now on that space I have an associated event space and I am going to say this probability is something that is going to assign a numbers event such that here not necessarily it could be full event space. So, I mean when I use the notation subset not necessarily it is a stick subset it could be entire set itself ok. I mean I mean if I want to specifically say that it is a proper subset then I am going to use this notation that means equality is not included ok. Otherwise if I just write it it will be just like this. Yeah, it says that this function p is basically function right it tells what is the value assigned to each possible event in omega. It says that for every element event will take a value which is between 0 1 and the entire sample space it is going to give a value of 1 and if you are going to take a sequence of mutually exclusive event the probability of their union is nothing but some of their probability. So, all of you understand what I mean by a sequence. So, you all might have heard about sequence of numbers right sequence of real numbers like x 1, x 2 where x 1 is a real number for example, x n can be let us say 1 by n. So, what is x n converges to if I set x n equals to 1 by n, but now I am talking about sequence of events right. What do I mean by sequence of events? It is just like indexing like I have set. So, what I mean by sequence means I have a indexed set of values right like you give me an number I mean a natural number that index is associated with some number. So, if I tell me i x i is the corresponding number if you tell me n x n is the corresponding. The same notion we are going to use for the sets like this for every possible index there is an associated set if you have such thing if you have such a sequence of this it is it is going to be just this ok. Now, and we are going to refer to p of e for any p of e we are going to just say is the probability of event ok. Now, let us see some of the things associated with this probability and let us say we can come up with some such probability function. Suppose, let us go back to our basic coin tossing problem. So, we have omega which consists of heads and tail, event can be either head or tail. So, let us say for if I say it is a fair coin what is that p of h you want to assign 1 by 2 right like both are equally likely. If let us say it is a biased coin then let us say somebody has tweaked the coin and probably likelihood of h is more than tail then maybe you want to assign probability of head to be something greater than 0.5. And now if I go to dice problem what is if I say it is a fair dice what is the probability you are going to assign to any number just 1 by 6 all of them are equally likely and if again, but that is not the necessary right. If you feel that you design such thing such that it could be biased. So, you may if you make some face heavy so that it comes more likely if the design is like that you are going to assign more than 1 by 6 further. And also note that this probability requires that p of s is equals to 1 that is it is all possibilities right you are going to say ok I am doing this event and these are the possible outcomes. When you do this experiment one of the things that has to happen right otherwise your definition of outcomes sample space was incorrect ok fine. You can it is up to you like you can define based on your model how on to you want to decide this p. So, you would like like if you want to model a fair to a fair coin you would like this p to be half for both the events and the same thing if you want to a fair dice you want to set p of i to be 1 by 6 for all i 1 to 6, but that is not necessary right this is up to you this is your function that you want to use it as whenever you want to model ok fine. Suppose let us say I have I say a fair coin and you keep tossing it keep on tossing it and after large number of toys you count how many heads you got and how many tails you got and divided by number of tosses you made what you expect that to be. So, if it is a fair coin you expect that to be close to 0.5 right. So, in a way you want to kind of interpret this probability as frequency of occurrence of that event when you do this experiment again and again ok. So, let us try to we will get back to this again ok fine. Now suppose let us say e and f are two events and I am interested in finding probability of e plus probability of f I am not saying that this e and f are disjoint ok they could have some common elements if that is the case what is the sum you are going to be like. So, one natural way to do this is either take probably. So, take it like probability of u f you want to do correction for this and that is going to be and why is this correction here they are not mutually exclusive right like suppose if you have a set A and another set B this set is entire this and this is B. If you are going to take this entire set once and you take this entire set once you would have added this region twice. So, you should have added I think. Yeah. So, you should have added instead of that because you need to find p e plus p r e plus. Instead of minus a there should be a positive. Oh right right ok. So, right. So, I mean just by this Venn diagram geometric interpretation you will see that fine when I did this I counted the common part once. So, I need to do it again because that has actually been added twice on the left hand side. Again this relation is fine it makes intuitive sense right the probability should be such that it should satisfy these conditions. So, now we have introduced this basic notion what is sample space, what is event and we have introduced this notion of this probability and it we want this probability should be such that it satisfies some intuitive properties. Ok, instead of going I want this property to satisfy something what we will now have to do is start with some basic assumptions on my probability space or my event space and define probability in a systematic way. So, that whatever the kind of things we want they actually naturally follow by our basic definition. Instead of we want this to happen you start with your basic frameworks such that they are induced to happen ok.