 Okay, students. Let me present to you moment ratios. Moment ratios, beta one and beta two. Beta one is connected with the concept of skewness. Beta two is connected with the concept of kurtosis. All right, let's begin with the concept of skewness. You know it very well that if the right tail of distribution is longer than the left tail, we say that the distribution is positively skewed. If the left tail is long, the right tail is skewed. Then we will say that it is negatively skewed. And if it is balanced, then we will say that it is symmetric, it is not skewed. So now to measure this, how will we proceed? Let me present it to you in a formal manner. Let X be a random variable with mean mu and variance sigma square, which obviously means that the standard deviation is sigma such that the third moment about mu exists. Third moment exists. Then the value of the ratio mu three over sigma cubed mu three over sigma cubed. This ratio is often used as a measure of skewness. Now focus on what we are talking about. We are saying that the third central moment, if it is divided by the third power of standard deviation, then this ratio can be used as a measure of skewness. This ratio will be negative if the distribution is negatively skewed. This ratio will be equal to zero if the distribution is not skewed, i.e. if it is symmetric and this ratio will be positive if the distribution is positively skewed. So it's a very simple way of assessing. Now if we square this ratio, i.e. the numerator will also be square, obviously, and the denominator. So now the thing will come, students, that is denoted by beta one and that may be called the first moment ratio and that also acts as a measure of skewness. So let's look at it carefully. Now we are going to square it. So now what do we have? We have mu three square over sigma cubed square. Now this sigma cube square, obviously you know that power multiplies in here. So sigma cube square means sigma raised to six because three twos are six. But then you can also write that sigma square cube because obviously if we do the sigma square cube, then two numbers are multiplying. Two multiplied by three is six and it is still sigma raised to six. So now when we write it like this, what do we have? The numerator is the same as mu three square and below we just said sigma square cube. But sigma square, the variance, that is the same thing as the second moment about the mean, the second central moment. So if we put it there, what does it look like? mu three square over mu two cube. This expression is denoted by beta one and this is called the first moment ratio. This is also a measure of skewness. Please don't forget one thing, since square is done, this quantity can never be negative. Just a little while ago, when square was not done, I told you that if negative answer comes, then it means the distribution is negatively skewed. If positive answer comes, then it means it is positively skewed. Here, since square is done, then it cannot be negative. So what will we get from this or what will we get from here? In a simple way, you will quickly understand that if this entity is equal to zero, that means that the distribution is not skewed. There is no skewness, there is zero skewness. So it is absolutely symmetric. But if this answer is zero instead of any positive answer, then that means that the distribution is skewed. Yes, but we cannot use this expression to determine its direction. Whether it is negatively skewed or positively skewed, but it is so known that it is skewed. After this, let us focus on the other concept, the concept of kurtises. You know that this is the concept of the amount of peakedness of the distribution. Is it too much peaked or is it like the normal distribution or is it even flatter than the normal distribution? So the moment ratio for this is beta 2. And what is the formula for beta 2? Beta 2 is given by mu 4 over sigma raised to 4. Fourth moment about the mean divided by the fourth power of the standard deviation. So obviously we can write this as mu 4 over sigma square whole square. And then sigma square because there is variance and variance means that, but that second central moment is also mu 2. So now we can write this expression as mu 4 over mu 2 square. And similarly it is written and it is called second moment ratio. And the notation for this, as I said, it is beta 2. So beta 2 is equal to mu 4 over mu 2 square. Beta 1, which I took a little while ago, was mu 3 square over mu 2 cube. And this one, mu 4 over mu 2 square. What is the interpretation of this? That is the phenomenon of peakedness. How will students expect this from this? This expression, if it comes out to be greater than 3, then we have, we conclude that the distribution is leptocratic. And you know what leptocratic means? More peaked, more peaked than the normal distribution. If this ratio comes out to be equal to 3, then we conclude that the distribution is mesokertic, meaning as much peaked as the normal distribution, that beautiful bell shaped curve. And if the ratio is less than 3, then we conclude that the distribution is platikertic. Even flatter than the normal distribution. So this is the story of moment ratios.