Introduction to Sigma Notation

Transcription

1 Itroductio to Siga Notatio Steph de Silva //207 What is siga otatio? is the capital Greek letter for the soud s I this case, it s just shorthad for su Siga otatio is what we use whe we have a series of ubers that eed to be sued It s that siple This is a really coo sceario i data sciece ad statistics, that s why the otatio exists You ay ot wat to be derivig equatios yourself, but it s really helpful to kow what soeoe else is talkig about There s a few quirks ad rules of otatio that are cooly used that are helpful to kow Here they are! What are the bits? Siga otatio is t just about the It coes with accoutreets that describe where the series to su starts, where it eds It also describes whether you wat a plai-vailla add the ubers up, or if you re goig to add toppigs - ultiply the ubers by soethig, square the, take soethig away etc The basic forat of siga otatio is this: There are alot of parts to this oe, let s defie the oe by oe: is siply the operator It s ot a variable or aythig you chage, it s a sybol like + or - It s short had for a whole lot of + sigs actually! s the series that will be sued There are a whole buch of xs ad they have a order To keep that order straight, they have subscripts (the little uber ext to the) i is the subscript of x, it s a covetio that gives each eber of the series x a ae, i this case i =, 2, 3, Here, is the last eber of the series It could be 0, 20, 00 or eve ifiity The betwee 3 ad just stads for all the ubers betwee the that are issig The piece of siga otatio we have above ca be expressed as a siple su I highly recoed that whe you are tryig to uderstad soethig, you siply write out the su to begi with Here s what that would look like here: = x + x 2 + x 3 + x + + x The idicates there are ebers of this series we have t writte out explicitly - i this case we do t really kow what is so we ca t write out the whole series Let s take soe cocrete exaples to ake this ore clear

2 If = 5 let s write out xi i log for xi x x 2 x 3 x x 5 = If = 2 let s write out yi i log for j= See what happeed here? We gave the variable a differet ae, y, the subscript y a differet ae, j, ad the ed poit of the series,, a ew ae too It all works i precisely the sae way: j= y i = y + y 2 Let s put soe ubers i to play We ow uderstad what all the bits are Let s see what happes with real ubers Let s say that x = 2,, 5, 6 All this is sayig is that s a series of ubers I this case x = 2, x 2 =, x xi = x + x 2 + x 3 + x x 3 = 5 ad = 6 If we wated to add the all together, we could say that where This could be writte out as: = = x + x 2 + x 3 + x = = 7 y =,, 0,, 0 y =, y 2 =, y 3 = 0, y =, y 5 = 0 yi = y + y 2 + y 3 + y + y 5 = 5 Let s try aother exaple Say that I this case If we wated to add the all together, we could say that where This could be writte out as: y i = y + y 2 + y 3 + y + y 5 Costats ca be treated specially i siga otatio Costats are just a uber that does ot chage, it is ot part of a series: it just lives o its ow Thik about this: xi ( + 2) = = 3 This is the sae thig as sayig

3 2 ( + 2) We have added to this suatio ties So we could siply rewrite this as = ( x + 2) + ( x 2 + 2) + ( x 3 + 2)+ +( x + 2) ( + 2) = 2 + I geeral, if a is a costat, the ( xi + a) = xi + a Let s cosider aother sceario where costats are iside a suatio of a series If we have (2 ), we could rewrite this as We just ultiplied each ter by 2 It would ea exactly the sae thig to pull this to the outside of the suatio like this (2 ) = 2 x + 2 x x x (2 ) = 2 ( ) I geeral, if a is a costat, the (a ) = a ( ) We eed soe exaples! Say that y =,, 0,, 0 I this case, what is (3 )? I geeral, whe a ed poit is ot specified, we assue we are to add the whole series So i this case, = 5

4 Let s try aother (3 ) If x = 2,, 5, 6, let s fid ( ) This is a little differet, but the sae priciples apply Here we will take = Sice is a costat, it ca be brought to the frot of the suatio like this: = 3 ( ) = 3( ) = 3(3) = 9 ( ) = = ( ) = (7) = 25 Cogratulatios, you just calculated your first average with siga otatio! We ca use siga otatio o ore tha oe series at the sae tie What if you have two series that iteract with each other? Suatio otatio is used alot i this cotext You ay see soethig like: y i x y i =, 2,, This siply eas that we ultiply each atchig ter of ad for all The equatio the looks like this: Let s take a cocrete exaple If =, x = 2,, 5, 6 ad y =,, 0,, 0 the we could fid xi y i Note how here We could t use because the series oly has four ubers i it! y i = x y + x 2 y 2 + x 3 y x y = = 5 x

5 Let s work this oe out: y i There are lots of uses for siga otatio Siga otatio is used all over statistics ad data sciece i differet fors You ll see ters squared, added, divided by ad so o If you keep these basic rules i id, you ll have a ore clear idea of what s goig o If you d like to see soe ore iforatio o siga otatio, try this site fro Colubia ( (C) Steph de Silva = 2() + () + 0(5) + 6() = = 2