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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.
Our focus in this part is on the function
f(x)=e-x2We wish to learn as much as possible about its graph. Please use algebra and Maple to answer the following questions:
Consider the initial Gaussian Distribution with x0 = 0 and you looked at earlier
Your answer should be accurate to 4 decimal places.
Since you have an improper integral, you must approximate the limit,
To demonstrate this, suppose we instead wanted to find
with the trapezoid rule to 2 places. Using the Maple command
> evalf(trapezoid(f(x),x=0..a,n));With 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 1.571The 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 .
Your goal in this section is to decide if/how the paramter x0 effects the integral of f(x).
Here, we consider the function from Part One,
and wish to find its integral from to . 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
by making the substitution (change of variable)
and your results from B. (your answer may have a parameter in it...)
Refer back to Part Two. From your earlier work,
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:
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.
Christine M Palmer