Understanding summation notation: (1/n) sum[i=1 to n] X_i

[Josh726]

So first off id like to quickly introduce myself. My name is josh, i enjoy learning and while im not in school or even college at this time i still like to teach myself different things, almost as a challenge.
Currently im learning about sigma. I have a pretty good understanding of it but came across a problem that has stumpped me. Im still trying to figure out the syntax for the equation to show properly so please bear with me
\(\displaystyle \displaystyle{1/n \sum_{i=1}^n X_i}\)

Now i know, or atleast i think i do, without anything left of the sigma it would be X₁ + X₂ + X₃.... + X_n so how exactly do i solve this problem?

 

[Deleted member 4993]

So first off id like to quickly introduce myself. My name is josh, i enjoy learning and while im not in school or even college at this time i still like to teach myself different things, almost as a challenge.
Currently im learning about sigma. I have a pretty good understanding of it but came across a problem that has stumpped me. Im still trying to figure out the syntax for the equation to show properly so please bear with me
\(\displaystyle \displaystyle{>1/n \sum_{i=1}^n X_i}\) ← please check for accuracy - does not make sense!!!

Now i know, or atleast i think i do, without anything left of the sigma it would be X₁ + X₂ + X₃.... + X_n so how exactly do i solve this problem?

What is the problem?

 

[Josh726]

If i am given a value for n say 10 what do i do with 1/n

 

[mmm4444bot]

If i am given a value for n say 10 what do i do with 1/n

Multiply the summation by 1/10.

The result will be the average of 10 numbers (represented by X₁ through X₁₀). :cool:

 

[JeffM]

So first off id like to quickly introduce myself. My name is josh, i enjoy learning and while im not in school or even college at this time i still like to teach myself different things, almost as a challenge.
Currently im learning about sigma. I have a pretty good understanding of it but came across a problem that has stumpped me. Im still trying to figure out the syntax for the equation to show properly so please bear with me
\(\displaystyle \displaystyle{1/n \sum_{i=1}^n X_i}\)

Now i know, or atleast i think i do, without anything left of the sigma it would be X₁ + X₂ + X₃.... + X_n so how exactly do i solve this problem?

This is a good question. It is not always clear what the sigma is adding up, but it is always true that the first item to the right of the sigma is included as is any item with an index that is the same as the index specified by the sigma. Once you have decided what the sigma applies to, you treat the sigma as a grouping symbol and apply PEMDAS.

So \(\displaystyle \dfrac{2}{7} * \displaystyle \sum_{j=1}^3x_j = \dfrac{2}{7} * (x_1 + x_2 + x_3) = \dfrac{2(x_1 + x_2 + x_3)}{7}.\)

In general,

\(\displaystyle \displaystyle a * \sum_{j=1}^nb_j \equiv \sum_{j=1}^nab_j \equiv ab_1 + ab_2 +\ ...\ ab_{n-1} + ab_n.\)

There are several common tricks involved in summation notation. Here is an important one for induction.

\(\displaystyle \displaystyle \sum_{j=1}^{k+1}u_j \equiv \left ( \sum_{j=1}^ku_j \right ) + u_{k+1}.\)