Lecture 18 : Decomposition of Distribution functions

Welcome to this lecture. Before we proceed with the discussions of this lecture, let's first quickly recall what we have been doing in this week. So in this week, we have looked at the measurable structure of random variables or random vectors and put them together with the domain side probability measure and when you put them together, we obtain the law or distribution of the corresponding random variable or random vector. Now using that law or the distribution, which is a probability measure on R, V, R in the case of random variables and on R, D with the Borel sigma field of R, D in the case of R, D valued random vectors. So using that probability measure, we have also defined the corresponding distribution functions. Now, what we have done in the previous lecture was that we looked at properties of such distribution functions. The thing to note is that we have defined distribution functions corresponding to probability measures on this Euclidean spaces together with the Borel sigma field. And then, for random variables or random vectors, we looked at the corresponding law and we said that the distribution function corresponding to the law is the distribution function of the random variable or random vector. And what we have seen is that the basic properties of probability measures immediately gives you the well-known properties of distribution function as you know about them, as you have studied in your basic probability course. So let us quickly recall these properties that we have already seen in the previous lecture and we can continue with the discussions of this lecture. So start with a random variable X. So just for simplicity, let us work in dimension 1. So this random variable, it is a real valued function on this probability space and it is a measurable function. So I look at its distribution function and what we have seen is that distribution function is a function defined on the real line and it takes values between 0 and 1. It is non-decreasing and right continuous function with some nice properties like the limit at infinity being 1, limit at minus infinity is 0, the left limit is exactly this, the probability of random variable X strictly less than small x and the right limit matches with the actual function value. So the function is right continuous and that is exactly probability of X less equals to small x. So here small x is any arbitrary real number. So once you have identified the left limits and right limits, the existence of those are confirmed and then only possible discontinuity that can happen for the distribution function is a jump discontinuity. And then you can also compute the size of the jump and that is exactly given by the difference of these values and that is computed as probability of the random variable X taking value the small x. So these are the properties we have seen and what we have also remarked in note 8 earlier that given probability measures, we can now transform it into a function. So from the collection of all probability measures on the real line with the Porel sigma field, we have obtained this function which is non-decreasing right continuous, limit at infinity is 1, limit at minus infinity is 0. So you have obtained the association from the collection of probability measures of that type of collection to the collection of functions of that type. So you have obtained the association. Now what we would do later on is to obtain function going in the opposite direction. So you would like to associate or make a association going in the opposite direction and what it is going to show is that given any function on the real line with values between 0 and 1 which is right continuous, non-decreasing, limit at infinity is 1 and limit at minus infinity is 0. So with these properties, if you get a function, then it will correspond to a probability measure mu on the real line together with the Porel sigma field. And then for that what will happen is that that given function will now correspond to the distribution function of that probability measure which you have just identified. So this we will do later on. Now recall that we have also mentioned this connection that if you start with the probability space and you are given the random variable, then you can construct the law. But we have also claimed that we shall construct association going in the opposite direction. So once you have completed that justification, that once you have constructed probability spaces and the random variable such that P compose X inverse is equals to mu for a given probability measure, then you can actually extend this identification with this general collection of functions. So what you will do? So you will first look at a general function with those nice properties. You first identify it as a distribution function of that probability measure and then once you have identified that probability space and the random variable, mu becomes P compose X inverse and that is what you have identified as the distribution function of that random variable. So we will do this connection once we have done the association going in the opposite direction. So this is what we are supposed to do. But for the purpose of this discussion, what we are going to assume is that this thing can be done, that this identification can be proved. So then any function satisfying these required properties, by that I mean that it is a function defined on the real line, takes values between 0 and 1, is non-decreasing, right continuous, limit at infinity is 1, limit at minus infinity is 0, then for such a function you can construct such probability measures and such random variables. Therefore such a function will become the distribution function of some random variable. So we are going to assume this fact in this lecture. So these are the basic facts about distribution function that we have already discussed. But now we are going to talk about other nice properties about the distribution function. But in the previous lecture, we have already mentioned that we are getting back the usual properties of the distribution function from the identification through the probability measures, right. So whatever properties you have already studied in your basic probability course about distribution functions, you are getting back and you are actually proving them through the properties of the probability measure in this course, right. So similarly you can try out this very well-known property of distribution function which you have seen in your basic probability course that the number of jumps of a distribution function is either finite in number or countably infinite in number. So please try to work this out. But then once you assume this fact that there are certain number of jumps of a distribution function, so since the number is finite or countably infinite, you can index the points where the jumps occur. So let's x subscript j denotes the enumeration of the jump points. So you just number them x1, x2, x3 and so on. So if it is finite, you will stop at a finite stage. If it is countably infinite, you will get an infinite sequence. So this is the list of jump points where the jumps are occurring. But then correspondingly we will also get the jump sizes, right, and let's call them as pj. So jump at the point xj is exactly given by pj, small pj. But then these points xj need not be in an increasing or decreasing order. So just be careful with that. It is just a list. It is just an enumeration of the points where the jumps are occurring. But now these pj's as identified earlier, these small pj's are nothing but the probability of capital X equals to small xj, right, that's the jump size and that quantity that is strictly positive because that's where the jumps are occurring. But then contribution from all the jumps if you add them together, that will be less or equal to 1. So this is a standard fact that you can always identify, that you can always prove. But here this sum that you are seeing, this sum is over the indices j and this refers to a sum involving finite number of terms in the case of finite number of jumps of the distribution function. Otherwise you will get countably infinite number of terms whenever you are dealing with infinite number of jumps for the distribution function, okay. So this is simply a basic fact whether the function, distribution function has finite number of jumps or countably infinite number of jumps, accordingly you will get a summation which is finite or countably infinite, okay. This is a simple fact. But then this familiar property for the distribution functions that you have just mentioned in this exercise, then it will also give you some ideas that the properties coming from the jumps, they should also follow, right. So this is the idea that once you have identified the basic properties of a distribution function from the properties of probability measures, then you are expecting that all the other well-known properties of distribution functions that you know about should also be proved, right. So one of the things is left as exercise there as the properties of the jumps, alright. So now let's look at more of these familiar properties and we are just recalling them. Look at this function fx superscript d. So what do we do? We define this function on the real line taking values between 0 and 1 and we look at this quantity defined on the right hand side. So for any point small x on the real line, I define this function as the summation value. So what is this? So you check whether the point small x is in this interval xj to infinity or not, okay. So if the point small x is in this interval, you will get the value 1. But this happens if and only if xj, the value xj is less or equals to x. So you are basically saying that you fix a point small x and look at all the jumps that have occurred before the point x. So you have looked at the real line up to and including the point small x and you are just looking at all the possible jumps that have occurred within that location, okay, within that interval minus infinity to x, small x closed, okay. So you are just looking at those jump points and corresponding to those things, you are just adding up the jump sizes. So that's exactly what you are doing in this summation, right. So this sum on the right hand side is exactly the sum of all the jumps that you find in this interval minus infinity to closed x. So that's how this function is defined. So this function fx superscript d simply looks at the number of jumps first in the interval minus infinity to x and then you simply add up the corresponding jump sizes. So it's just the contribution from the jumps. So that's what you are looking at in fx superscript d. But then there are some very nice properties about this function that you have just obtained from the contribution from the jumps. So this function is non-decreasing, it is right continuous, limit at minus infinity is 0 and limit at infinity is exactly the contribution from all the jumps. So remember we have mentioned that given a distribution function, the contribution from sum of all these jumps is less or equal to 1. So this quantity that you get, this quantity is less or equal to 1. So the limit at infinity is something less or equal to 1. So you now see that the fx superscript d which is purely considered from the jumps contributions, this is a non-decreasing function, it's right continuous and you can also identify the limits at infinity and minus infinity. So these are very standard facts that you can prove easily and this you must have seen in your basic probability course. If not, please take this as an exercise, please work them out. But now let's focus on that limit at infinity. So I just look at that summation or the contribution from all the jumps, all the jump sizes. So now this quantity as I have remarked that this is less or equal to 1. So call that limit as alpha. So this is just a shorthand notation that is going to be used now. So I am just looking at the contribution from all the jumps. So that's why I call it as alpha. Now if fx is continuous, that means fx does not have a jump, so there is no contribution from the jumps, then alpha must be 0. And accordingly you will also have no contribution from the jumps anywhere. So fx superscript d, that function which we have just considered will also be identically 0. So if the distribution function is continuous throughout the real line, then both these quantities alpha and the function fx superscript d must be 0. So fx superscript d is a function which becomes identically 0 if the original distribution function is continuous. But then what happens if fx has at least one jump? So you just move away from continuity, allow at least one jump. Then there will be some contribution coming from the jumps and that will be contained in that quantity alpha. So now alpha will be between 0 and 1, 0 excluded. So there will be some non-trivial contribution from this at least one jump quantity. And now what you can choose to look at is this scaling factor. So alpha is now strictly positive. So you can now look at 1 by alpha, divide the actual contribution from the jump, so divide it by alpha. So fx superscript d divided by alpha. So look at that function. So that function has these properties following from the properties of fx superscript d. So what are these properties? Since alpha is strictly positive, the 1 by alpha fx superscript d that function will remain a non-decreasing right continuous function with limit at minus infinity being 0, limit at infinity being 1. So just to repeat, you are looking at this scaled version of fx superscript d and it turns out to be non-decreasing right continuous with some nice limits at minus infinity and plus infinity. The limit value being 0 and 1 respectively. So therefore what you can immediately now claim is that 1 by alpha fx superscript d in this case when there is at least one jump is a genuine distribution function of some random variable. So here we are just making that connection from note 11 which we have mentioned at the start of the discussion. So we are saying that if you have a function with all these nice properties, it must be the distribution function of some random variable. Again just to recall, we have mentioned in note 11 that we are going to prove this fact later on. But for now we are going to assume this as a fact. So this is the contribution from the jumps and we have identified this thing as a distribution function when alpha is something non-trivial, when alpha is strictly positive. But then you can now choose to look at the remaining portion that takes away the jump parts. So you take away all the contribution from the jumps. So you are looking at the distribution function, given distribution function and subtracting out the contribution from the jumps. And whatever you get, call it fx superscript c. So that is now a function which is difference of two function values. So therefore it is a real valued function taking some real numbers as its possible values. So fx superscript c is defined as the original function, original distribution function, but you are taking out all the contribution from the jumps. But then again this will also be some familiar properties to you. So I am just recalling them for the sake of the record. Please work out these exercises if you have not seen this before. So fx superscript c has some nice properties. So first of all we have mentioned that fx superscript c is a function defined on the real line and takes real values. But then the first property itself says that it actually takes values between 0 and 1 and moreover the fx superscript c is dominated from above by the actual distribution function. So in particular fx superscript c, it is a non-negative function and it takes value at most 1. So that is the first property. So fx superscript c now becomes a function from the real line to the interval 0, 1, the closed interval 0, 1. But then it has also some other nice properties like its non-decreasing and its continuous. So fx superscript c is non-decreasing and its continuous and you can also identify its limit at minus infinity which is 0. So please try to work out these properties of the function fx superscript c. But then you will be wondering about what happens at infinity. So fx superscript c is a non-decreasing function. So the limit at infinity will exist. So call that limit as beta. So remember for the contribution from the jumps, we have looked at fx superscript d. There we looked at the limit at infinity, we called it alpha. Here we are looking at the remaining portion. We are taking away all the contribution from the jumps. We have constructed this function fx superscript c. It turned out to be non-decreasing. I am now looking at the limit at infinity and calling it beta. But now since fx superscript c is bounded between 0 and 1, this quantity, this limit, this beta should be between 0 and 1. So that is not a problem. So it follows easily. But then recalling the fact that the contribution from the jumps, that fx superscript d has a limit at infinity to be alpha. So that is basically sum of all the jump sizes. Then you can make some nice comments about alpha and beta. When you put alpha and beta together, you can make some very nice comments. So if beta is 0, then what you are saying is that this limit at infinity for the fx superscript c is 0. But then this is the one of the bounds, upper bounds for the fx superscript c. So therefore what you will now identify that there should be no contribution from this part. So beta is 0 if and only if fx superscript c is identically 0. And this can only happen if fx, the original distribution function, is identically equal to the contribution from the jumps. So fx superscript d. So this can only happen if and only if beta equal to 0. So that is exactly what the first statement says. So what basically is said now is that all the contributions are coming from the jump parts. So that is the first statement. So the second statement says, assume alpha is 0. So take alpha is 0. So that means there is no contribution from the jumps. So it is exact opposite situation compared to the Riemann one. But this can only happen if and only if there is no contribution from the jump. So therefore this function fx superscript d is identically 0. But then as per definition, fx superscript c must be the difference of fx minus fx superscript d. Therefore fx superscript c under this condition must be exactly be fx, the original distribution function. So alpha is equal to 0 if and only if there is no contribution from the jumps and fx superscript c equals the original distribution function fx. So these are two extreme cases, alpha equal to 0 or beta equal to 0. But then if beta has some non-trivial contribution, if beta is strictly positive, then you can now scale this fx superscript c by 1 by beta. If you recall, we have done a similar argument involving fx superscript d. So we looked at the limit value alpha, if the alpha was strictly positive, if there was some non-trivial contribution from the jumps, then we scaled it, scaled the function fx superscript d by the alpha and we observed some nice properties. We do the same here. If beta is positive, look at 1 by beta, the scaled version of fx superscript c. I say that this is now a distribution function of some random variable. Why? Again, let's try to check whether fx superscript c satisfies all those nice properties. So remember fx superscript c is taking values between 0 and beta, beta is upper bound for fx superscript c. So therefore 1 by beta of fx superscript c takes values between 0 and 1. That's the first point. Next is fx superscript c is non-decreasing and beta is positive. So therefore 1 by beta fx superscript c is still non-decreasing. You can also show by a similar argument that it is also continuous and you can also identify the limits at plus infinity and minus infinity as 1 and 0 respectively. So it satisfies all the required properties and therefore by the discussion in note 11 earlier, this must be the distribution function of some random variable. So that's the third statement. And then put alpha and beta together, what you can now try to show is that alpha plus beta actually equals to 1. So therefore alpha basically looks at contribution from all the jumps, beta looks at the remaining part. What we are saying is that put them together, you are going to get the total quantity is exactly equal to 1. Please try to check this. So with that identification in mind, what we are now going to do is to look at certain types of distribution functions. So what we have observed so far is that the original distribution function fx has a part coming from the jumps and there is a remaining part which is continuous and we have proved some nice properties about these separate parts and with this gives us some motivation for looking at the definition 4 and this tells us that we can look at certain special type of distribution functions. So the first thing that we are looking at here is a distribution function which can be written in this form. So remember this was the form that we have looked at when we looked at the contribution from the jumps only. So you are just looking at all the jumps that occur within point x and then you just add up the corresponding jump sizes, so all the jumps that have occurred up to that point. So that's how that jump contribution was considered. But if a distribution function is purely of this type for some finite or countably infinite set of xj's with this pj's being non-negative and sum up to 1, then you are going to say that f is a discrete distribution function. So it's purely made up of jumps. So without loss of generality you can always assume pj is strictly positive, it's greater than 0, you can ignore the equality case because if pj is 0 then there will be no contribution coming from this summation. So therefore without loss of generality you can also take pj to be greater than 0, you can ignore the equality case. But just for simplicity let's continue with these assumptions. So in this case we are looking at a discrete distribution function which is defined as the made up of purely of jumps. So the total contribution from the jumps must be 1 and the distribution function has exactly this form made up of some kind of a combination of indicators for some appropriate finite or a countably infinite set of xj's. So that's where exactly the jumps are occurring for that distribution function. So this is what a discrete distribution function means and the exact counterpart for the continuous case is this. So if you get that the distribution function f is continuous or not then you say that f is a continuous distribution function. So there is no contribution from the jumps, there are no jumps, so you just get a purely continuous function and this is the case you are going to say that f, the distribution function is a continuous distribution function. So this is a very simple definition. But then depending on these two types of distribution functions we are now going to obtain certain nice classes of random variables. So this is what we are going to discuss later on. But now continue with these notations that we have discussed in note 14. So just to recall, we have considered the contribution from the jumps, we have defined the fx superscript d function. That function has a limit at infinity which we denoted as alpha. Then we got rid of all the contribution from the jumps, we looked at the remaining part and that we called as fx superscript c and that was also non-decreasing function. So we looked at the limit at infinity called it beta. So remember all those alpha and beta. So remember if alpha is 0 then there is some non-trivial contribution from jumps. So then you can try to show that 1 by alpha fx superscript d is a discrete distribution function. So you have already mentioned this fact that 1 by alpha fx superscript d in this case when alpha is positive, in this case it's a distribution function. Now you can try to check that it's exactly of the relevant form and it's a discrete distribution function. Similarly, you get a statement for the continuous part. So you look at beta positive, so there is a non-trivial contribution outside the jumps. So there is some contribution coming from continuous parts. So in this case you can try to show that 1 by beta fx superscript c which we already identified as some distribution function, this as a continuous distribution function. So please try to write down this proofs, it's a very simple argument you just have to verify the corresponding structures. But then with this identification at hand what do you get? You get that given any distribution function fx of a random variable x, we have this convex linear combination. So remember in the case when alpha is positive and beta is also positive, consider that case then 1 by alpha fx superscript d, this part within the brackets, this is a genuine distribution function and it's a discrete distribution function. Look at this part 1 by beta fx superscript c, this is also a genuine distribution function and it is a continuous distribution function. So the terms, the functions within brackets are distribution functions but they have been scaled according to alpha and beta. So these scalars that I have put outside this alpha and beta, they are non-negative and they add up to 1. So remember all those properties. So here on the right hand side you have a convex linear combination of distribution functions. Alpha and beta being the scalars for the convex linear combination. But then what you have basically you are saying is that given a distribution function you can split it as a convex linear combination of a discrete distribution function and a continuous distribution function. But you will ask what happens to this convex linear combination if alpha equal to 0 or beta equal to 0. Then of course you cannot divide them by these 0 quantities. So if alpha is 0, 1 by alpha does not make sense. But then ignore this first term there. Remember if alpha equals to 0 then beta must be 1 and then fx is exactly equal to fx superscript c. So this equality still makes sense when you ignore the contribution coming from the jumps as alpha equals to 0. So ignore this first term. So it is just rewriting fx equals to fx superscript c as beta times 1 by beta fx superscript c. Exact counterpart of this is the case when beta is 0. So then again ignore this contribution coming from the continuous part. So there is no contribution in fact. So ignore this term here. So purely contributions are coming from the jumps. So fx is exactly equal to fx superscript d. But in this case alpha equals to 1 and then what you are just rewriting it as fx equals to fx superscript d equals to alpha times 1 by alpha fx superscript d. So this equality still makes sense as long as you interpret it right. So you can still continue to say that fx itself becomes a convex linear combination of distribution functions in all of these situations. That alpha equals to 0 or beta equals to 0 or alpha and beta both are positive and adds up to 1. But then you can now ask is the decomposition that you have obtained is this unique? And the answer to that is yes. So why? Because fx when you are writing it, if you can write it in terms of such a convex linear combination with gamma delta non-negative adding up to 1 for f1 which is a discrete distribution function, f2 which is a continuous distribution function, then you can show that this must be unique. So if you can write a general distribution function as a convex linear combination over discrete and continuous distribution functions, then this identification is unique. So that's the exercise. Please try to prove this. In particular, the identification that we have done in the previous page splitting it in terms of the appropriate contributions coming from the terms the fx superscript d and the fx superscript c, that was one example of a convex linear combination. But this exercise is saying that such a combination must be unique. So if you combine these two things together, so on one hand in note 16, we have the existence of such a combination and exercise 5 says such a combination if it is there, then it must be unique. So put them together, we have a complete description of the decomposition into discrete and continuous parts. So given a distribution function, you can split it. So either it's purely discrete or purely continuous or it's a mix of this with a explicit convex linear combination coming from the jump parts and the remaining part, the continuous part. But if you look at the continuous parts separately, a further decomposition is known. We are going to discuss that part in week 8 later on. So we do not go into this. We will require much more measure theoretic setup before we can discuss about this further decomposition of the continuous part. We will come back to this. So there remember, we have looked at the fx superscript d function, which is purely made up of the jump parts. And with appropriate scaling, we can get it to be a discrete distribution function. Now with that at hand, we are now going to consider certain class of random variables, which exactly correspond to this kind of distribution functions. So what are these? A random variable X is said to be discrete, if there exists a finite or a countable infinite set S, which is a subset of the real line, such that P compose X inverse, the law of X associates full mass to this. So the probability that the random variable takes values in this set is 1. So remember, any singleton set is a Borel subset of the real line. So therefore, if you consider finite unions or countable unions of those, you will get any finite or countable infinite set like S as a Borel subset of the real line. And P compose X inverse, which is defined on the Borel sigma field of R can therefore associate certain values, certain weight, certain size to this set S. And we are saying that that size must be the full quantity and that is 1. So if this happens, then you say your random variable X is discrete. Now we are just recalling your basic definition of a discrete random variable that you have seen in basic probability courses. But then we are using this measure theoretic knowledge and putting the definition in terms of the corresponding law. And you will see the advantage of this later on. But then, since S is finite or countably infinite, you can enumerate the elements in the set S. So if it is finite, you will get x1, x2, xn up to a finite number n. And if it is countably infinite, you will get a sequence of points. So just enumerate them. Then using this countable additivity of the probability measure P compose X inverse, look at this quantity. So this is as per the definition for a discrete random variable, P compose X inverse of the set S is exactly equal to 1. But then the set S is a countable disjoint union of these singletons. So split it according to the countable additivity of the probability measure P compose X inverse. Then what you get is this summation over all the points on the set S. Just rewrite the summation using the notations that we already learnt about and it is nothing but the addition of all the probabilities that X equal to Xj. So we add up all the things where Xj falls in that set S. So this is a relation that sum of all these probabilities adds up to 1. So this is the relation star which we shall recall in a few minutes once more. But remember since P is a probability measure, P compose X inverse is a probability measure, these quantities, this contribution coming from the jumps here, this must be non-negative. So you are just looking at addition of certain non-negative quantities and they must be 1. So that is what this star relation suggests. But without loss of generality, you can only consider the points Xj when these quantities are positive, when these quantities are not 0. So without loss of generality, ignore those terms for which the Xj gives you 0 probability. So probability X equal to Xj equal to 0, such terms we will ignore. It is not a big deal. They will not contribute to the summation here in the star relation. So without loss of generality, this set S can be considered to be the set of points Xj where the probability associated is positive, where there is a genuine jump. So the set S basically is collecting these jump points. So whenever there is a jump, you just collect it in the set S. And for the case of these discrete random variables, you get this set S and we are going to call it as the support of this random variable X. And just as you expect, whenever you will look outside this set S, if your point small x is outside, then the probability associated to there must be 0 because there is no jump there. You have listed all possible jumps in the set S. Just note this that if S is finite, you can now rearrange these Xj's so that you can get an increasing order for all these points. So if you have finitely many points, x1 up to xn, you can easily sort them and put them in an increasing order. So this is a basic observation. This you can use it sometimes. But you be careful whenever S is countably infinite, sometimes it may not be possible to rearrange all these jump points. So let's ignore that for now. So now let's go back to the question about the law of these discrete random variables. So now what we are going to look at is this quantity P compost X inverse associated to general sets. So for any set A on the Borel sigma field, if A is a Borel subset, it's there. So you look at this quantity. But then if P compost X inverse associates full mass to the set S, then P compost X inverse of A is nothing but P compost X inverse of A intersection with the set S. So this is an easily verifiable property. So please check this. But then you split this according to the usual countable relativity that we have used a few minutes back. So S, the set S, the support is made up of all these singleton sets where the jumps are occurring. So you write it as a summation over xj's. So that is nothing but probability of x belonging to this set. But then you rewrite it using this notation. So you just check whether the point xj is in the set or not. So if the point xj is not in the set, then this intersection is empty, there will be no contribution. If this point xj is in the set, then this intersection is exactly the singleton set xj. So in either case what you can do, you can write it as this quantity, this contribution coming from the jump multiplied by the Dirac measure. So this is what we are just rewriting. You just observe that in either of the cases, when xj is in the set A or not, you can write it using this notation. So therefore you just rewrite this expression in terms of the Dirac measures. But remember we use this notation pj's for the jump sizes and just rewrite it using this pj's. So therefore what we have started off with, p compose x inverse of A is nothing but this summation over xj's and it is this summation over pj's, then delta of xj of A. So Dirac measure, then you are considering certain combinations. But remember you are considering a discrete random variable. So these pj's are strictly positive whenever there is a non-trivial jump, you are looking at that. So that is the jump size. It is non-trivial, positive quantity and they sum up to 1. So therefore this quantity, whatever this is, this is a convex linear combination of the weightages coming from the Dirac mass. And hence what you will immediately obtain is that p compose x inverse is exactly this convex linear combination of the Dirac masses. Remember these are the similar things we have discussed in week 2. So when we looked at measures, we said that certain convex combinations of probability measures also gives you probability measures. Okay, great. So just to recall, just to repeat, p compose x inverse of arbitrary set A can be written as this convex combination where pj's are exactly the jump sizes and delta xj of A are exactly the weightages coming from the Dirac masses. So that is why p compose x inverse exactly turns out to be of this form. So whenever you get a discrete random variable, you get this quantity, great. Now you can also connect it with discrete distribution functions. So if x is a discrete random variable, then its distribution function must be discrete. And this is a if and only if condition that if its distribution function is discrete, then the original distribution function x must be discrete. This is a if and only if condition. And in this case what you can see is that between any two jump points, so if there are no jumps in between two such jump points, then fx will remain a constant. So if you plot the graph of the distribution function, it will remain flat between the consecutive jump points. So some final comments that for the case of discrete random variables, we have now connected the measure, measure theoretic structures with the setup from the basic probability. So we have proved these properties of distribution functions, identified jumps, we looked at discrete distribution functions, looked at discrete random variables, but we commented that we will come back to the continuous cases later on. But we can now therefore use the standard analysis that you do in basic probability. In particular, since you have already identified everything from the discrete random variable, you can now go back to the probability mass functions and talk about all the relevant properties. And then just to make this comment about continuous distribution functions. So remember we have associated the discrete random variables with discrete distribution functions. But then you will ask what happens to the continuous distribution functions. Does it correspond to some types of random variables? And what happens if the distribution function is a convex linear combination of discrete and continuous distribution functions? So in all of these cases, you can identify appropriate class of random variables. So we will do that identification and we will identify these other types of random variables and that discussions will be done later on when we identify more structures about continuous distribution functions in week 8. So there we are going to talk about continuous random variables. So in the next lecture, we will come back to this issue about random variables whether you can construct them according to specified law. That discussion we will do in the next lecture. We stop here.