 Dear students, let me present to you the CDF of the discrete uniform distribution and let me do this with the help of an example. Suppose that we are rolling one fair die one time and we want to have the CDF of the distribution in front of us in the mathematical form, algebraic form and also we would like to draw the graph of the CDF. All right, so how do we start? Subsepele pehla column X, what are the possible values of X? It represents the number of dots on the uppermost face of the die when we roll it. So then obviously X's values they will be 1, 2, 3, 4, 5 and 6. After that second column probabilities since it is a fair die therefore 1 by 6, 1 by 6 so on all of them 1 by 6. Now for the CDF cumulative distribution function, what do we have to do? We have to construct one more column and this is the column of cumulative probabilities. Here I avoid writing capital F of X because it has a reason which I will share with you later. In this place its heading I would like to write cumulative probability, so what is its procedure? Just like you do cumulative frequency, the first value as it is and then the result plus the next value and then the result plus the next exactly we will do here. So the first probability as it is 1 by 6 and then 1 by 6 plus 1 by 6 gives us 2 by 6, 2 by 6 plus 1 by 6 gives us 3 by 6 and we go on like that. So the last one is 6 by 6, i.e. 1. Now these cumulative probabilities have come, we have to use them in order to write the CDF in its proper form. What you now see on the screen is very very important and I would like to request you to pay a lot of attention to it. Take a look at the first value that is written there that is 0 and the last value that is 1. Note that it would always start from 0 and go up to 1. The second thing is to note that the portion of the x axis written against 0 that starts from minus infinity and note that the portion of 1 against x axis that ends at plus infinity. So here the message for you is that whenever you write the CDF, you have to exhaust the entire x axis. It should always always start from minus infinity and go up to plus infinity. All right, if this is clear, then look carefully at it. Now I would like to tell you to look at the cumulative probability written below 0. When we did that step a little while ago, we said first value as it is, then you can see that below 0 that first value is written. 1 by 6, below that 2 by 6, 3 by 6 and so on until we get 6 by 6, i.e. 1. After this, note that the 1 by 6, 1 by 6 that is written against the x axis, which portion is written in front of it. Note that we are writing 1 less than or equal to x less than 2. It is very, very important that you note that the 1st value has equal sign and the 2nd value has no equal sign. The 2 by 6 below that is 2 less than or equal to x less than 3 and the next one is 3 less than or equal to x less than 4 and so on until we get 6 less than or equal to x less than infinity. Now I will tell you why this is. Now you will say that I left the first 0 deliberately and now obviously I will come towards it because I have to think about that part as importantly as any other part. Now when we wrote 1 by 6, 1 less than or equal to x, then the one above that is 1. We cannot include 1 in that because 1 is included in the lower part. So the one above that will be exactly as you see on the screen, minus infinity less than x less than 1. This is how it is completed, everything is covered up to minus infinity to infinity. Before I tell you the reasons which I want to tell you. I would like you to look at the graph of this particular cumulative distribution function. Do you not see that it is a beautiful staircase type graph where the steps are rising by equal height and also the length or the width of the step all of them they are the same except the first one which starts from minus infinity and the last one which goes up to plus infinity. So it is a beautiful step function, a technical term here, it is a step function and from the basic rules pertaining to discrete probability distributions, you already know that the graph of the CDF of any discrete distribution would be a step function. It would start from level 0. It would rise up to level 1, but the important thing is to note that the more symmetry and beauty is seen in the graph of the discrete uniform distribution of the CDF, the rest will not be like this, the rest will be that the height will not be the jump at any other place and similarly its weight that can also be different. But for the discrete uniform distribution, it takes this beautiful pattern. Now the last thing that I would like to explain which must be in your mind is that thing about putting the equal sign before the X and putting not putting the equal sign after the X. We have to consider the probability of getting one on the uppermost face of the die is equal to 1 by 6 and after that when we said that the probability of getting two on the uppermost face of the die is also 1 by 6, so the concept of cumulative is that what is the probability of getting either a one or a two when we throw this die? What is the probability of getting either a one or a two if we throw this die only one time? Well, students, that probability is 2 by 6 obviously, because if we look at it from the classical definition itself that there are six possible faces, two of them are favourable that either one comes or two comes, so then that favourable over the total 2 by 6, the CDF in front of us is 2 by 6 against the value of X which is written as 2 less than equal to X less than 3, so that 2 is in it, but 3 is not in it because if this question is asked what is the probability of getting either a one or a two or a three when throwing one die, so its answer is 3 by 6, so the next 3 by 6 is written against the values of X, where 3 less than equal to X is in it, so this is the fundamental way of interpreting any CDF, so if you focus on it carefully, then you will realise that this is exactly the correct way of writing it, if I write it against the values of X, then I am saying that the probability of getting either a two or a one is 1 by 6, so that is not correct, 2 or 1, 1 or 2 is the probability of getting either a one or a two or a two, so the probability of getting either a one or a two or a two is 2 by 6, so the two by 6 is written as 1 by 6, so the probability of getting either a one or a two or a two is 1 by 6, so the probability of getting either a one or a two or a two is 1 by 6, so that is not correct, it is 1 by 6, that X will be anywhere between 1 and 1.99999, the jump that it takes is on 2, it does not take 2 from the beginning, in its graphical version, now I will tell you in the end that whatever I said, look at it carefully and focus on it, try to understand this with reference to the graph, i.e. both complement each other, the graph does not take place unless you write this and it understands better if you look at the graph, all the best.