 Dear students, let me present to you the mean and variance of the discrete uniform distribution. Let us derive the formula for the mean in the case of the uniform 1n distribution, a uniform 1n distribution concept when we have x values as 1, 2, 3, so on up to n and the probability attached with each one of them as 1 over n. Alright, let us derive mu, the mean is the expected value of x and because it is a discrete situation, therefore the expected value of x is given by the summation of x into p of x. Now, p of x we will substitute 1 over n. So what do we get? We get summation x into 1 over n. Now after this, we take 1 over n out of summation because it is a constant and now what we have become? We get 1 over n into summation of x. If we open it and write summation x, then what will we get? 1 over n plus 1 plus 2 plus so on up to n. So now you know a basic rule that the sum of the first n natural numbers is given by n into n plus 1 over 2. So therefore, mu, the mean is equal to 1 over n into n into n plus 1 over 2. So obviously, n will cancel with n and we are left with mu is equal to n plus 1 over 2. Here, we derive mathematical expression of the mean of the discrete uniform distribution that we denote by capital U and then a bracket in which we write 1 comma n. After this, let us try to derive the variance of the same distribution. Variance ka shortcut formula kya hai? You know that it is expected value of x square minus expected value of x whole square. So, what is that equal to? Obviously, that is equal to summation x square into p of x. Now, p of x is equal to 1 over n. So, let us substitute. So, what do we get? Summation x square into 1 over n. So, what do we get? 1 over n summation x square. If we open it like this, then it will be 1 over n and what is multiplied with it? 1 square plus 2 square plus 3 square plus 1 up to n square. But again from school days, we know that the sum of the squares of the first n natural numbers is given by n into n plus 1 into 2 n plus 1 over 6. So, then obviously, n will cancel with n and we are left with n plus 1 into 2 plus 1 over 6. Abhi ye variance nahi hai? This is simply the expected value of x square. Ab jo shortcut formula hai variance ka uske andar hum ye cheezain dal dete hai. So, as you can now see on the screen, the variance which is also known as sigma square, denoted by sigma square is equal to e of x square minus e of x whole square, i.e. n plus 1 into 2 n plus 1 over 6 minus n plus 1 over 2 whole square. Uske baad students obviously, you can open the bracket, you can square, you can do all that is required and the derivation is in front of you. And what is the final result? Well, just before the final result, what do we get? We have the expression 2 into n plus 1 into n minus 1 over 24. So, 2 by 24 to zahir hai ke 1 over 12 ke brahbar hai. Upar jo n plus 1 into n minus 1 hai, what is that equal to? Of course, it is equal to n square minus 1 square is liye ke hum jantin hai ke a plus b into a minus b is equal to a square minus b square. To ye jo n square minus 1 square hai, yani n square minus 1, this is being divided by 12 and this, my dear students, is the expression for the variance of the discrete uniform distribution. Agar hum iska square root lein to what do we get? We get the standard deviation of the discrete uniform distribution. Before I close this discussion, I would like to give you an example by which you can appreciate some of what I have presented. Let us get back to the mean, the mean, the simple thing. Dekte hai ek example ke zeh liye ke kya us ka mean vaakai n plus 1 over 2 ke brahbar hai. So, let us consider the rolling of a fair die. Agar ek fair die ko hum roll karein to kya x ki values hongi when x is representing the number of dots on the uppermost face of the die. Of course, the values will be 1, 2, 3, 4, 5 and 6. So, you can see that in this particular example, n is equal to 6. Acha. Iske baad isko visualise karein. Har ek ke saath probability kya attach hui hai? It is 1 by 6 because it is a fair die. To agar iska graph hum draw karein to x axis kya upar? 1, 2, 3, 4, 5, 6 aur har ek kya upar ek vertical line and the height of each one of them is 1 by 6. Ab students zara khaur ki je, 1, 2, 3 aur pheer 4, 5, 6. Ye 3 aur 4 ke beech me, 3.5. 3.5 je point hai x axis kya upar, waha pe aap mirror place karein to aap dekh rahe hain. Ke the left hand side of the distribution is the mirror image of the right hand side. To pheer aap ko pata hai ke iska matlab hai ke absolutely symmetric distribution around the point 3.5. To pheer aap jante hain ke jab symmetric ho, to mean is exactly at that central point. So, this means that graphically we have judged that the mean of this particular distribution is 3.5. To abhi jo formula hum ne derived kya kya uske tehad bhi, are we getting 3.5? Well, what is the formula? Mu is equal to n plus 1 over 2. If n is equal to 7, I am sorry, if n is equal to 6, what do we get? 6 plus 1 over 2, 7 by 2, yani 3.5. So, jo hum ne graphically jaj kya liya, the formula is exactly giving us that same result. These are the algebraic expressions that I have just derived for you for the mean and variance of that particular uniform distribution that is written as capital U 1 comma n.