 Dear students, I'm going to present to you a very interesting concept, and that is the concept of a degenerate random variable. A degenerate distribution is a probability distribution, the support of which is only one single point. Now, let me say it again. A degenerate distribution is a probability distribution with support on only one single point. Let's talk about normal distribution, the most well-known distribution in the world. When you hear a normal distribution, what comes to your mind? You think of that bell-shaped curve. On the top, you can see the bell shape. But the question is, what do you say about the x-axis below? Obviously, you will say that the entire x-axis is involved, from minus infinity to plus infinity. The entire x-axis is on top of the bell-shaped curve, which we call a normal curve. It is drawn on top of that and it is on top of that. So that is the case when, which is just the opposite, I should say, of what I just said. That is the case in which the entire x-axis is acting as the support of your distribution. And what I just said is that it is such a distribution that the x-axis below is just one point above it. Such a distribution is called a degenerate distribution and such a random variable is called a degenerate random variable. So let me try to give you one or two examples. Suppose that I pull out a coin and interestingly, we find that on both sides of that coin, we have a head. Now if I toss it, what am I going to get? Obviously, no matter what I do, when I toss it, I will get a head. Another example. Suppose that I pull out one and we find that on each one of the six faces of that die, we have that same number. Then you find three dots. Then obviously, when you will toss it, so note that you are performing a random experiment. So that is why it is said that this kind of a distribution satisfies the definition of random variable, even though it does not appear to be random in the everyday sense of the word. Therefore, it is called degenerate. Well, degenerate actually, this is also the way to understand that the x-axis, as if it collapsed, it became only one point. And all the things we have are on one point. For example, what I just told you three, that there are three dots on each side of that die. This means that the number three on the x-axis, your distribution is on top of it. It is a vertical line, the height of which is one. Why? Because the height will denote the probability, so if it is one, what does that mean? The probability is one that you will get three if you toss this particular die. What does probability one mean? Don't you know that one is the probability of a short event, cent per cent, hundred over hundred, one. So this is short energy. If there are three dots on each side of it, it is a short event that you're going to get a three. So degenerate, the normal curve or any other distribution is the first one. This is your support, it collapses, it is only one single point. In this particular example, it is three. After this, note that if we formally call it using algebra, we will call it like this, that if the space of the random variable contains only one point k, for which the probability of p of k is greater than zero, then p of k has to be equal to one. Now, as I just explained, the mean of that random variable is equal to k, that is, mu is equal to k, and the variance of that random variable is equal to zero. Now, consider a third example. Suppose that a teacher in a classroom administers a test, and the test is out of ten. Now, in that classroom, suppose that twenty children are sitting. If every single one of those children gets seven marks out of ten, all the seven marks of the children come, then it is not obvious that the mean mark is also seven. Because if you add the number of seven twenty times and divide it by twenty, then 140 over 20, that will again give you seven. So, the number of each one is the same mean. That is what I just said, that mu will be equal to k. And the second thing is variance is equal to zero. Look, everyone has a number of seven. There is no variation whatsoever. There is no variation. So, how can variance be anything other than zero? So, as I said, it's a very, very interesting concept, the concept of degenerate random variable.