 Now, let us try to make this more formal ok. Now, how to define probability of a random variable on a point and probability of a random variable on a set. Let us take a point x belongs to r ok. So, again always recall that x is a map from omega to r. So, range is the entire real line ok. So, I can always ask the question what is that what is that x is going to take a value x small x ok. So, just again just think that this is your real line all the way from minus infinity to plus infinity and this is your sample space omega. What x is doing is is putting every point in this sample space to a point on the real line. Now, for this I may ask ok if I take a particular point what is the probability of taking this particular point on my real line. So, that is like ok what is x equals to x what does this mean? This is all omegas in capital omega my sample space such that x of omega is that small x ok. So, if I am going to say this is my point it may happen that multiple points in this sample space could get mapped to the same x right. I just showed you the example here right. If I want x equals to 3 both these 1 2 and 2 1 they are getting mapped to the same x equals to 3 ok. So, I am just saying that all those points in capital omega which will get mapped to that point x you are interested in and that is naturally a subset of capital omega. And now instead of a point you may want to ask take this particular region what is the probability that x is going to lie in this region and A is that region. Now, if you want to ask what is the probability that x is in that region this is set of all those omegas such that x of omega belongs to that subset that set A that is all maybe like this bunch of points here they are all go to some points in this region and all these points is what those omega this this part is capturing and that is another subset of omega. Now, this is simple notation we are going to say now we can ask we know that x equals to x is some event if I if it is as event I can ask the probability of that event. Now, probability that x equals to x I usually simply denote it is in this way P with a subscript denoting that random variable and inside the point x that I am and if I am interested in knowing this x belonging to some subset A I will just write like this ok. If you understood this just let us quickly do this examples let us x is a random variable which denotes the sum of the outcomes sum of the outcomes right. Now, what is P of what is the probability that my random variable x takes value 5 ok 4 by 36 or 1 by 9 everybody agrees or anybody has different answer. So, 5 can happen in how many ways 1 4 4 1 2 3 3 2 and now what is the probability that my x takes value in the set 4 and 5 1 by 7 by. See now I am asking x to take either value 4 or 5 right you already know it takes 5 in 4 ways and how many ways it can take value 4 3 ways right 2 2 3 1 1 3. So, already 4 as their additional 3. So, this is a value. So, now, see that that now random variables we have, but that random variable is just a phase for us now like what we have done is omega and the events we have now kept them aside now we are defining things in terms of the random variables, but they are connected to those all events. Now, I am basically now saying ok I have the random variable given to me I am asking the probability of that random variable taking some value, but that is all coming from the basic events itself. Probability x equals to x means that probability x equals to x means that means that is corresponding to some event. Once I know that event I know the probability of that event. So, in that way when I am asking some random variable it is connected to the events and on that events I know probabilities that is indirectly I am saying that ok I will now define probabilities on the random variables itself. Now, there are different types of random variables we can think of depending on the type of range called this as the discrete and random variables. So, whenever I mean we have two kind of things obviously, two classification can happen discrete random variables and continuous random variables. Whenever the values taken are finite or countable we are going to call the discrete random variables and we have already seen many discrete random variables right. The sum of the outcomes of rolling a 2 dice in that case x can take value 2, 3 all the way up to 12. So, this is in fact like it is going to take finite values. If you are interested in a random variable where you are going to toss till head appears. So, in that case x can go from 1, 2, 3, 4, 5, 6 all the way up to infinity. Here it is infinity, but still countable. So, we are going to treat it as a discrete random variable. On the other side if it does not belong to either case we are going to call it as a continuous random variable. That means, there are uncountably many points values my random variable can take ok. So, for example, if I am interested in temperature of a room temperature of the room if let us say I am interested in degree Celsius and it could be anything between let us say 0 to 40 all value between 0 to 40 can happen ok. So, it is a continuous things here and similarly height of a person may be your height varies from between 50 centimeters to 200 centimeters. Any value in this range is possible it is a continuous ok like that and you will be given some minion max range. If it is going to take all possible values in that range we are going to call it as a continuous random variable. Now, based on what we have defined now we want to aggregate start aggregating this information ok. Random variable give you a nice way of representing translating everything is on the numbers. Now, on those numbers you may want to start asking questions whether my random variable is going to take value less than this or more than this or it is going to be in this interval or it is not going to be in this interval you can ask all these questions ok. They all correspond to different events, but now instead of talking about events we will just talk about random variables ok. The first thing we are going to look into is something called cumulative density function CDF. So, CDF is again a function which is a map from real numbers to interval 0 1. Notice that its range is entire real line and its domain is 0 1 interval ok. So, if you have a random variable x and I want to compute its CDF at a given point x that is defined as probability of x taking the value between minus infinity to do x which is same as saying that probability that x is going to take value less than or equals to small x. So, what CDF is saying is and this should be this is for all x belongs to R. So, CDF of a random variable x at point small x is nothing, but that random probability of that random variable taking value less than or equals to that value small x ok. An example ok, let us say I have a random variable x which takes only 3 values 1, 2, 3 and the probabilities of their probabilities I have given to you. Probability that x takes value 1 is half, probability that it takes value 2 is 1 by 3 and the value it takes value 3 is given 1 by 6. Now, we want to find out its cumulative density sorry cumulative density function. Let us plot how does it look for this particular example. So, I have to plot like x is on the x axis and y is my f of x. So, now, let us look into till point 0. Let us take I mean I have to define it for all possible x right. Let us start define let us looking let us start looking into what happens at x equals to 0. What is its value? So, that is basically I am asking what is the probability that my value random variable x takes value less than or equals to 0. Is it taking anything less than or equals to 0? No right because it is only taking value 1, 2, 3. So, cumulation that point is going to be 0. So, in fact I should have such this. So, it is going to remain 0 till this point and now if I going to take any point between 0 to 1 it is still going to be remain 0 and when I include this point 1 when I made this x equals to 1 then this half will come into picture. In the sum probability x is less than or equals to a x when x is 1 that can happen with probability half it can take 1 right and that is included. So, there is a jump of amount half way. But now between 1 and 2 can my random variable x take any value between 1 and 2? No right because it is only taking 1, 2, 3 here. So, till 2 this is going to remain flat and at 2 it is again going to make a jump and like that when 3 it is going to make a jump and after that it is going to stay at 1 it cannot go beyond 1 ok. So, this is like graphical way another way of representing the same thing is you be more descriptive let us say x is less than equals to 1 it is 0 between 1 and 2 it is half between 2 and strictly less than 3 it is 5 by 6 and for x greater than or equals to 3 it is going to be 1 and a percent like this ok. Now because of this definition itself CDF has certain properties the very natural property that emerges is f of x is not decreasing in x everybody agree with this everybody see it has to be non decreasing I am saying non decreasing instead of increasing because it can remain flat ok that is why they are saying non decreasing can remain flat at some points. And now if I let x goes to infinity what does f of x of x when x goes to infinity yeah this is because when I am letting x here to infinity x is I am basically letting x I am interested in letting x take any value that means I am interested in the whole space outcome space. So, that is why it is going to be 1 on the other hand when I let x go to minus infinity I am letting x to be minus infinity here I want x to be less than or equals to minus infinity that means I am basically not considering any possible outcomes of x right that means it becomes null set and it becomes a 0 and that is why it is 0. Another property that emerges is something called right continuity ok. All of you know what is a continuous function or anyone here who do not know what is a continuous function ok all of you know what is a right continuous function anyone here who do not know what is a right continuous function yeah you do not know ok. So, fine I mean you are the only one who do not know talk to me later. And now yeah I mean all this straight forward properties right nothing I mean you these are straight forward and you actually go and actually prove them even though these are all pretty intuitive in this case you can just go and apply the axioms we have and demonstrate that they are all indeed true. Now actually now probabilities like ultimately CDF is capturing probabilities it is basically cumulative accumulating this probabilities. Now we can ask the questions like what is the probability that x is going to take a value between x and x is between x and y. This is nothing but probability that x is going to take value less than or equals to y and minus probability that x is less than or equals to x. We like just think of like this like I have this and I want to ask the question what is the probability that my random variable is going to take value between this. So, one thing is you consider all the values below this and from this you subtract the value below x then you will exactly get this portion and that is what it is doing. So, take everything between y and now take everything below x you subtract this we will get and but this is nothing but f of y minus f of x as per our definition. So, this probability is nothing once we know the CDF and you want to know the probability that your random variable lies between those two points all you need to do is take the difference of your CDF at those two points ok and our actual definition of the probability here if you notice f of x is x less than or equals to minus x sorry x less than or equals to x. But then what if I want to compute what is the probability that x is strictly less than x you can compute it using the limiting notion you just take f of x minus h and let h goes to 0 from the top side. So, I mean you are approaching h in a positive direction and compute it and you will get it and since f of x is defined for all possibilities this is fine and f of x and probability that x less than or equals to x need not be equals to probability that x equals to x this need not be the case ok. So, just let us see the example let us say this is a 1, 2 and 3 here and there is a jump here. Now in this case let us say there is a jump of half here. So, in this case probability that x is less than or equals to 1 is half. Now what is the probability that x is less than 0.9 0 what is the probability that x is less than let us say 0.999. So, what is the probability that x is strictly less than 1 it is still going to be 0, but x less than or equals to 1 it is going to be half ok. So, notice that here this amount of the jumps are actually corresponding to the probability of those points. So, here the jump is like half that corresponds to probability that x equals to 1 and here the jump this much of jump is corresponding to 1 by 3 and that is the probability of x equals to 2 and here the jump is 1 by 6 and that is the probability that x is going to equals to 3.