Special Summations
Gauss Sum
Newton noticed the following pattern.
101*100 = 10,100, but he counts everything twice
The answer is 10,100/2 = 5050
Jose M. Fernandez
Tuesdays and Thursdays from 11am - 12:15pm
Office hours: Tuesdays 1:30pm - 3:300pm BS 159 of by appointment.
(Alternative, but more rigorous) Wooldridge, “Introductory Econometrics”, 4th or 5th edition, South-Western. 2008.
Second, download Rstudio (the free open access version ) from http://www.rstudio.com
Formal Definition:
Informal Definition:
| Type of Assignment | Percentage |
|---|---|
| Computer Assignments | 20% |
| Quizzes | 20% |
| Midterm | 15% |
| Final | 25% |
| Project | 20% |
This class requires a commitment both to attendance and participation.
The textbook makes an excellent reference, but makes a poor substitute for attendance.
Ask Questions
When something is even the slightest bit unclear (and especially when you are completely confused!). Please do not be afraid to interrupt my lecture for further clarification.
Start your paper early!
Let \(x\) be a random variable.
Take \(n\) draws from this distribution.
The sum of these draws is given by
\[ x_{1}+x_{2}+ . . . + x_{n}=\sum_{i}^n x_{i} \]
The mean or average is defined as
\[ \bar x=\frac{1}{n}\sum_{i}^n x_{i} \]
The variance is defined as the average squared deviation from the mean.
Population variance \[ \sigma^2 = \frac{1}{n}\sum_{i=1}^n (x_{i}-\mu)^2 \]
Sample variance
\[ s^2 = \frac{1}{n-1}\sum_{i=1}^n (x_{i}-\bar x)^2 \]
Geometric Series
\[ \sum_{i=1}^n b^i = \frac{1-b^n}{1-b} \]
Gauss Sum
One day, Newton was annoying his 1st grade teacher. She decided to give him a task to keep him busy.
“Sum all the numbers between 1 and 100.” Confident this would keep the young boy busy she continued with her lecture, but a minute later Newton was once again disturbing class.
How did he do it?
Gauss Sum
Newton noticed the following pattern.
101*100 = 10,100, but he counts everything twice
The answer is 10,100/2 = 5050
Gauss Sum: Check the formula by adding the numbers from 1 to 4.
\[ \sum_{j=1}^J j = \frac{(J+1)J}{2} \]
If x and y are random variables, then
\[ \sum_{k=1}^n (x_{k} + y_{k}) = \sum_{k=1}^n x_{k} + \sum_{k=1}^n y_{k}\]
if \(\alpha\) is a constant, then
\[ \sum_{k=1}^n \alpha x_{k} = \alpha \sum_{k=1}^n x_{k} \]
A review of Basic Statistics
If you need a math refresher look at my notes on blackboard.