Subsections

• Purpose
• A. The Geometry of Gaussian Distributions
• B. Definite Integrals of Gaussian Distributions
• C. Effect of the parameter x₀
• D. Effect of the Parameter $\sigma$
• E. Putting It all Together
• Conclusion

Improper Integrals

Purpose

The purpose of this lab is to acquaint you with the family of functions known as Gaussian distributions. The two aspects of them we wish to become familiar with are their geometry and various integrals of them.

A. The Geometry of Gaussian Distributions

Our focus in this part is on the function

f(x)=e^-x2

We wish to learn as much as possible about its graph. Please use algebra and Maple to answer the following questions:

1.

Does it have any symmetry?

2.

Does it have any asymptotes?

3.

Does it have any local extrema? If so, where?

4.

Does it have any point(s) of inflection? If so, where?

Your write up should support all of your answers.
We now generalize the function a bit by adding a parameter, x₀. The function now becomes

f(x)=e^-(x-x₀)2

5.

So if x₀ = 0, it is the same as in part A. Describe the effect of the parameter x₀ on graph of f(x).

In this section, we add a second parameter, $\sigma$, and seek its effect. The function now becomes

$$f(x)=e^{\frac {-(x-x_0)^2}{2 \sigma ^2}}$$

6.

To see what its effect is, try simultaneously graphing the function for $\sigma = 1,2, and \: 3$.

7.

Next ,use calculus to see what are the point(s) of inflection of the new function. Your answer here should be consistent with the graphs you just found.

8.

At this point, you should be able to conclude what the effect $\sigma$ has on the graph; please describe it.

B. Definite Integrals of Gaussian Distributions

Consider the initial Gaussian Distribution with x₀ = 0 and $\sigma = 0$ you looked at earlier

f(x)=e^-x2

1.

Your goal here is to apply knowledge of integration to obtain a value for the improper integral

$$\int_{- \infty}^{\infty} e^{-x^2} dx$$

Your answer should be accurate to 4 decimal places.

Note that this integral cannot be done using antiderivatives and the Fundamental Theorem of Calculus. The function simply does not have an antiderivative! (try finding one!) Thus the only option you have is a numerical method (your choice). Note: take advantage of symmetry and save some work!

Since you have an improper integral, you must approximate the limit,

$$\lim_{a \rightarrow \infty} e^{-x^2} dx$$

To demonstrate this, suppose we instead wanted to find

$$\frac{1}{1+x^2} dx$$

with the trapezoid rule to 2 places. Using the Maple command

  > evalf(trapezoid(f(x),x=0..a,n));

With $f(x) = \frac{1}{1+x^2}$ and varying both a and n, one gets the following table of values:

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
a 		 n 		 evalf(trapezoid(...) 		 a 		 n 		 evalf(..) 

100 		 20 		 2.186 		 600 		 800 		 1.568 

100 		 100 		 1.567 		 600 		 1000 		 1.569 

100 		 300 		 1.561 		1000 		 1000 		 1.576 

100 		 600 		 1.561 		 1000 		 2000		 1.570 

200 		 600 		 1.566 		  1000 		 4000 		 1.570 

200 		 800 		 1.566 		 4000 		 4000 		 1.576

400 		 800		 1.568 		 4000 		 8000 		 1.571 

400 		 800 		 1.568 		 4000 		 10,000 		 1.571 

 		 		 		 8000 		 10,000 		 1.572 

 		 		 		 8000 		 15,000 		 1.571 

 		 		 		 8000 		 20,000 		 1.571

The method here being for a given value of a to take n large enough so that no change is seen to 2 decimal places. Then we increase a and again increase n enough so no change to 2 decimal places is seen. When a is large enough that no change, to 2 places, is seen between two consecutive values of a, then we are done. In the above example, which converges pretty slowly, this happens for a around 4000. We take the value of the integral to be 1.57 to 2 decimal places. Note that we save three places at all times and observe normal roundoff rules (1.568 rounds to 1.57; 1.561 rounds to 1.56 etc). The exact answer is $\frac{\pi}{2} = 1.570796$.

2.

Suppose we wished to scale the function f(x) so that the area under the curve from $- \infty$ to $\infty$ was one. What value of k would you use to do this if $$f(x) = ke^{-x^2}$$

C. Effect of the parameter x₀

Your goal in this section is to decide if/how the paramter x₀ effects the integral of f(x).

1.

What is the integral of $$f(x) =e^{-(x-x_0)^2}$$ from $\infty$ to $\infty$?

Instead of doing this numerically, as in A, lets make a change of variable and make use of the result from A. Make the substitution (change of variable) u=x-x₀ and determine the integral.

2.

How would you scale it so the integral would become one?

D. Effect of the Parameter $\sigma$

Here, we consider the function from Part One,

$$f(x)=e^{\frac {-(x-x_0)^2}{2 \sigma ^2}}$$

and wish to find its integral from $- \infty$ to $\infty$. Again, instead of doing this numerically, we can make use of earlier results with a change of variable. beginenumerate [1.] Determine the value of the integral

$$\int_{- \infty}^{\infty} e^{\frac{-(x-x_0)^2}{2 \sigma ^2}} dx$$

by making the substitution (change of variable)

$$u= \frac{x-x_0}{2 \sigma ^2}$$

and your results from B. (your answer may have a parameter in it...)

E. Putting It all Together

1.

The goal here is simple: pick a constant k so that the function

$$\int_{- \infty}^{\infty} ke^{\frac{-(x-x_0)^2}{2 \sigma ^2}} dx = 1$$

Refer back to Part Two. From your earlier work,

2.

What vertical line is it symmetric about?

3.

Where are its points of inflection?

4.

What does its graph look like?

Conclusion

The family of functions you have studied in this lab have many applications in mathematics, science, and engineering because many naturally occuring processes generate data which follows them. Such processes include SAT scores, heights of people, the strength of bolts made on an assembly line, the percentage of people voting for a candidate based upon a sample, and many more. They have a number of different names you may encounter:

$$* Gaussian \: distribution$$

$$* bell \: curve$$

$$* normal \: distribution$$

and were originally developed by the mathematician/physicist Karl Gauss (1777-1855) for use with astronomical data he collected. The functions are noteworthy as their areas involve improper integrals which cannot be done with antiderivatives.