 Hello friends in this session. We are going to discuss something about submission notation. You would have seen this notation used extensively in statistics and functions and matrices and other places So let us, you know, give you give you a little Overview of what exactly this notation is and how it properties how its properties are like So What is summation location notation you have come across this sign sigma So it is a Greek Greek letter. It's it's called Sigma. It's capital Sigma in Greek and You know, it is shorthand for addition of a sequence of any kind of numbers Okay, so let us say if if you pay attention to this particular sequence, you know, these are x1 x2 x3 x4 Let us say there are these many xn numbers. So there are let's say n numbers and numbers and you want to Use them in your literature or let's say well while communicating So it will be very cumbersome to write x1 x2 x3 x4 and so on. Let's say n is thousand So to write thousand variables like this will become very cumbersome So hence we use a shorthand and this is a notation for that. So I'll also explain sideways. So if you see in this Example one where x1 x2 x3 to xn are numbers in a sequence and then that is No, that is, you know notified as Sigma sign like this Sigma and There is this letter I which is the counter or we call it as index as well. So I goes from one and Ends up at n What is it mean? What is it? What does it mean when I say I goes from one to one? So I the value of I will change Incrementally, so I will be first one Then to add one to one then add one to two you'll get three then for then five likewise till I value becomes n Right and then this is X I so what would it mean? It means that you have to start from X and instead of I first I you write one which is this one Okay, then add a plus sign then write X and then the second Subscript this part is called sub subscript. This is subscript Okay, so the second subscript will be two right and then again add write Three and keep on doing this till you add all the X's X is till Xn, right? An Example will make it more clear. So let us say in this second example Before that few definitions. So I is index of summation. You can see that I is equal to one is the lower limit X I is the variable and I is equal to n is the upper limit, okay? So let us see this in example two. So if you see I is varying from seven to ten Now why I that means first value of I will be seven then add to it why eight that means I value has become eight Then it is why nine and then finally why ten because you will not write why eleven because I maximum upper limit is ten Okay, so this is what example two says example three if you see I is equal to one to I is equal to n summation I Okay, summation I means the value of I will keep on changing like first I value is one So I have written one then the second term will be to the next I third the third value of I is three then four then five Likewise till n minus one and finally n. So this is some of first n natural numbers Now come to example four here. It is I equals to three to I equals to six So you might be thinking why is I not starting from one? So I can have any value starting value till it is lesser than the upper limit, right? But upper limit has to be more than lower limit now if you this is a summation I square that means First value of I is three. So three squared and next value of I is four Squared five squared and finally six squared you'll not be writing seven squared because the upper limit of I is six Now, let us understand some properties of summation operator So here I have mentioned I is equal to one to n summation C X I where C is a constant C is a constant and X I is a variable, right? So the idea or the left-hand side of this particular the equation says that it is C X 1 plus C X 2 plus C X 3 So on and so forth till C X n So what can you do? You can take C common and you can write X 1 plus X 2 plus so on and so forth till X and Now this is C times nothing, but what is X 1 X 2 X 3 X n? We just saw above so this is nothing but X I where I is varying from 1 to n Okay, I is equal to n Example for this would be let us say you have 2 X I and I is equal to let's say 1 to 10 So it will be nothing but two times summation X I I is equal to 1 to 10 Okay, first one. Now, let us understand this particular thing Example so here are two variables X and Y's and they are varying. So hence LHS of that will give you what? So one I first value is one so X 1 Y 1 Then it goes to X 2 plus Y 2 then X 3 plus Y 3 and so on so forth till X n plus Y n Now you can rearrange them to write as X 1 plus X 2 all X together X n plus Y 1 plus Y 2 plus All Y's together right now the first this term this term is nothing but Summation X I and I is varying from where to where 1 to I is equal to n and The second term is nothing but summation Y I and I is varying from 1 to that Now third one also follows. So instead of plus there is minus. So here it will become minus simply Right, so second and third are quite similar. So what do we learn? So if there is Summation on two variables together so we can split it into two summations first on the first variable and Second on the second variable Similarly, if there is difference with the summation on difference of two variables, then that is equal to Summation of first one minus summation of the second variable Yeah, there are a few more properties of summation operator But that those will be more relevant in that's a topic under functions and matrices and other things So we'll stop here. These three properties are good enough for you to Understand the concepts of statistics basic statistics of elementary level. Thank you You