 Dear students, once one has studied all the different rules that pertain to probability distributions, discrete probability distributions, continuous probability distributions, univariate distributions, bivariate distributions, after having had a broad idea about this wonderful discipline and this wonderful subject, one can begin to focus on some particular well-known distributions. There are many, many of them, but some distributions which are fundamental or important and they find applications in real-life situations and they are well-known, they are then generally considered one by one after having had a study of the overall rules and various theorems, etc. of probability distributions. What I am going to present to you now is the Discrete Uniform Distribution, which is one of the simplest possible distributions that you can think of and which actually has application in real-life situations. The Discrete Uniform Distribution, alright, what is a Discrete Uniform Distribution? Students, it is a symmetric probability distribution, a symmetric distribution here, just may a finite number of values are equally likely to be observed. In other words, if the discrete random variable X can assume n distinct values and every one of these n values has equal probability 1 over n, then we are dealing with a discrete uniform distribution. Let me give you some examples to explain what I mean. The simplest example is that of the case when we are going to toss a coin, the number of heads that we may get, we are tossing one fair coin only one time and X represents the number of heads that we might have. Students, it is a very simple distribution, you know that a discrete distribution is in which we have just two columns, a column of X and a column of probabilities of X values or probabilities of sum that has to be 1, as you can now see on the screen, capital X to a variable, it contains just two values, 0 and 1, because obviously if you throw a coin once, either it will have a tail, which means X is equal to 0, number of heads is equal to 0, or it will have a head, which means X is equal to 1, number of heads is equal to 1, when you are tossing it only once, then two heads cannot come, right? If you had tossed twice, then obviously X could also have been equal to 2, now let's stick to this, just one toss, so X's values are 0 and 1, and I said it is a fair coin If it is a fair coin, obviously the probability of getting a head is half and the probability of getting a tail is also half, so the second column is written as half and half, obviously the sum is 1, so this is a perfect example of a discrete uniform distribution, note that the word here is uniform, why am I saying that it is uniform? Because of the fact that both of the probabilities are equal, if it is half and half instead of 1, 1 by 4 or 3 by 4, then of course it was still a discrete distribution, but not a discrete uniform distribution, the values you will attach with it only then, when all the probabilities are 2 and if they are more than 2, then more than 2, all those probabilities are equal and the sum of them is 1, so in order to explain this point let me give you another example, let X denote the number that we get on the uppermost face if we roll a fair dice one time, Ludo's dice are one, and roll one by one, so how did X define? The number that we get on the uppermost face, so the first column, X column, what will we write in it? Obviously 1, 2, 3, 4, 5, 6, because you know that there are 1 and 2, 6, now since it is a fair dice, so each one of these faces is equally likely to occur, so if each one of them is equally likely to occur, it is obvious that for each one of them the probability is 1 by 6, and you are doing this 6 times, when you add it, you get 6 by 6 and that is equal to 1, so this is another perfect example of a discrete uniform distribution, now you will know even more clearly that you have 6 probabilities but each one of them is the same, so the love of uniform will be attached only when we have this kind of a situation, if we make this graph, then what will be the shape of the graph? In the first example, on X axis, you take 2 points, 0 and 1 and on that you have to draw a vertical line on 0 and on 1, and the vertical lines you will draw, their height will be 1 by 2. In the second example, you will mark 6 points on the X axis, in this case 0 will not be there and we will mark, that is 0 will be there but we will not use it for our distribution and we will mark and use for our distribution the 6 values, the 6 points on the X axis, 1, 2, 3, 4, 5 and 6, each one will draw a vertical line and each one will draw a vertical line, what will be its height? 1 by 6, so as you can now see on the screen, don't you see that if all of them are of the same height, so if we put our hands on it, then it is going horizontal, so this is exactly why we call it a discrete uniform distribution. This is the basic concept of discrete uniform distribution.