 Dear students, let me present to you the concept of the mean, variance and standard deviation of a random variable. First let us consider the case when the random variable x is a discrete variable. So in this case, the expected value of x will be given by summation x into p of x where x, of course, are the values, x values are the values of the random variable and p of x represents the probabilities associated with those values. So, the expected value of x is equal to summation x into p of x. Let us look at this in detail. Suppose that these values, the values of x, suppose that they are a1, a2, a3, and so on. Then the formula that I just presented to you, summation x into p of x. A1 multiplied by p of a1 plus a2 multiplied by p of a2 plus a3 into p of a3 plus so on, so on. This sum of products is seen to be a kind of a weighted average of the values of a1, a2, a3 and the weight associated with each ai is p of ai. The probabilities they are acting as the weights. So, in general, we say that arithmetic mean is simply add the values and divide by their number, summation x over n. But if you open it a little bit and write it, what will happen? It will be something like this. If the values are once again a1, a2, so on, up to an. So, what we say that summation x over n, this will actually be a1 plus a2 plus a3, so on, so on, plus an over n. Now, this n in the denominator, if you attach it to each ai separately, what will it be? a1 over n plus a2 over n plus so on, so on, an over n. If we say this, 1 over n multiplied by a1 plus 1 over n multiplied by a2 plus so on, so on, 1 over n multiplied by an. And if we reverse the order of this, what will it be? a1 into 1 over n plus a2 into 1 over n plus so on, so on, an into 1 over n. So, this thing that is being multiplied by 1 over n, this is, in this case, you can see that only one quantity is being multiplied. So, this is the case that each one of those ai's is being given the same weight. They are all attached to the same weight, 1 by n, 1 by n, 1 by n. Generally, it does not have to be the same weight. So, a b pehle aapke saamne rakha, ai into p of ai. In general situation me yeh hota hai, ke wo jo probability hai, that acts as the weight. If all the values are equi probable, then we will have something like what I just said. But if the various values are not equi probable, then they will, each one of those values will have a different weight. To yeh is kusum ka concept hai, bahar hal, it is that same mean, which we are talking about all the time, the most fundamental concept for any random variable. Is ke baad next, let us talk about the variance. So, variance kya jizein? Again, let us consider a discrete random variable, where support a1, a2, a3, yani jo values hai us variable kyi, yahi hai. Aur uske baad, aab agar hum variance nikal na chata hai, to uska formula kya hoega? As you know, the variance is given by the expected value of x minus mu, mu whole square. Abhi thodi der pehle jizeen ka zikar ho raha tha, mean ka jo zikar ho raha tha. Of course, uske ko mu se denote krtein hai. So, expected value of x minus mu whole square jo hai, in case of a discrete variable, that will be equal to summation x minus mu whole square into p of x. To isko bhi agar hum ussi tarah se khol na hai, to kya baneega? a1 minus mu whole square into p of a1, plus a2 minus mu whole square into p of a2, plus so on, so on. So, therefore, this particular sum of products will also be interpreted as a kind of a weighted average of the squares of the deviations of the numbers a1, a2, a3 from the mean mu. And again, the weight attached with each of these expressions, a1 minus mu whole square will be p of a1, the probability. Ye ho gaya variance aur iske baad, we all know that if we take the square root of the variance, then we get what is called the standard deviation. And as you know, the standard deviation is one of the most important measures of dispersion. Iske lawa shortcut formulae ki bhi baad, yehi pa kar lete hain, yeh jo variance ka formula abhi aapke saamne rakha, through a very, very simple derivation, as what is now in front of you on the screen, we can show that the variance is equal to the expected value of x square minus the expected value of x whole square. In the other words, the variance is equal to e of x square minus mu square. This is a shortcut formula, jab aapne compute karna hota hain, yeh derive karna hota hain. To yeh jo formula hain, yeh aksar baat handy hota hain, you get your result quickly, and that is why it is called the shortcut formula.