IC - Project 3Pa

Numeric integration& Gaussian distribution : 

Numeric integration& Gaussian distribution Angel Vera Lepe. A01063075 Omar Jair Casillas Cerón. A01062990 Larry Zanmar Sánchez Castelán. A01062991 Karen Monserrat Rodríguez Solís. A00343622

Intro : 

Intro So far we’ve learned that to calculate the area under a graph we can divide the area in several rectangles and then calculate the area of all of them together. We must know that there’s also another way to calculate an area, by using trapezoids instead of rectangles. This info is important because in this project, we use a special formula named “Simpson’s method” which combines both rectangle (by 1/3) and trapezoid (by 2/3) calculations to get to a more exact result.

Slide 3: 

Turns out that to accomplish this project, we were given an excel file which has the development of a function using the Simpson’s method with an interval of [6, -6]. Also there’s a graph of the function and its anti-derivative. Remember that there are an infinite number of anti-derivative functions (+C), so the anti-derivative function shown on the graph for the specific limits of integration you entered is the one that always begins equal to zero.

File : 

Here’s a little view of what we were given. File Function Anti-derivative

1st part: : 

Using the excel file we have to experiment using 3 different functions, and compare our results and our graphs: Check out on the next images from our excel files that the intervals used are different. This is because that way the graphs are easier to understand and we don’t get messed up by asymptotes. 1st part:

Slide 6: 

Function Integral

Slide 7: 

Function Integral

Slide 8: 

Function Integral

2nd part: : 

The original function we were given turned out to be: 2nd part: And its interesting because it describes a distribution from negative to positive infinity, but it doesn’t have an elementary anti-derivative equation. The shape it makes is called "Gaussian distribution."

Slide 10: 

The total area under this curve equals one (100% probability). The center represents the "normal" outcome of an event (the mean), and you see that the function has its highest value in the center. But other outcomes (other values of x) can occur with some probability. Our job here is to find out: Probability of an event (x) occurring within one, two and three standard deviation (-1,+1), (-2,+2), (-3,+3)of the mean. Probability that an event will occur in the "tail" more than one, two and three standard deviation higher than the mean. Probability of an event occurring in either tail outside of "six sigma" . Distance from the mean [-a,+a] which captures 25% 50%, 75%, 90%, 95% and 99% of all outcomes.

Slide 11: 

Results:

Slide 12: 

Now if we all have to know, In probability theory and statistics, the standard deviation of a statistical population, a data set, or a probability distribution is the square root of its variance. Standard deviation is a widely used measure of the variability or dispersion, being algebraically more tractable though practically less robust than the expected deviation or average absolute deviation. It shows how much variation there is from the "average" (mean). A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values.

3rd part: : 

Her we had to ask 100people who belongs to the ITESM “community” their height, to analyze if it vary as a Gaussian distribution around a mean value. To do that we accomplished this steps: Create a frequency of occurrence histogram for each measure. Calculate the mean of your distribution. Calculate the standard deviation of your distribution. Compare your histogram with the general shape of the Gaussian distribution. What percentage of your population is within +/- one, +/- two and +/- three standard deviation of the mean? 3rd part:

Slide 14: 

Slide 15: 

To calculate the mean, we just divided the sum of all the heights over the number of heights asked: To calculate the standard deviation, we used the next formula: Our result was: 11.198

Slide 16: 

Our graph clearly isn’t even close to look like the Gaussian distribution.

Slide 17: 

The percentages are this: Its funny that there’s actually one person who deviates more than three standard deviations or the mean.

The End : 

So this is all we had to do. We have accomplished this other project. Thanks for your attention. The End

Co-evaluation : 

Karen – 1 Angel – 1 Larry – 1 Omar – 1 Co-evaluation