MME 523

Analysis for Applications

September 30,2003

Areas and Bell Shaped Curves

As discussed in class in the past, the function is at the basis of the bell shaped curves which are central to much of statistics. The goal of this week is to study them both numerically and analytically.

Part One: Basic Properties of :

Please show, either by hand or using Maple,

• its graph
• that this function is symmetric (aka “even”)
• find its points of inflection

Next we are interested in finding a value for the integral

As pointed out earlier, the Fundamental Theorem of Calculus is no help here because there is no antiderivative. Only numerical estimates may be used.

First start by coming up with a definition for what the symbol

simply means!

Using numerical methods from last week, as well as what you know about the function, please find an estimate of the integral

to 5 decimal places. Document all steps and methods used to achieve this. Your work should be consistent with the definition you came up with.

Now define a function proportional to which is a probability density function.

Part Two Integral Tables For Your Function.

To four decimal places, please find values for the integral of your density function for x from 0 to b where b is equal to

.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5 and 4.0

In each case indicate how many trapezoids were needed and what sort of analysis you did to ensure that 4 decimal place accuracy was achieved. Summarize your work in table form.

Part Three A Better Algorithm?

Please find out about and summarize Simpson’s Rule for next week. How does it compare conceptually and in performance to the Trapezoidal Rule? What is it based upon? Can you find an error estimate? Please comment on itt.

Pick two examples of computations that you did with the Trapezoidal Rule and your probability density function and see how comparable computations with Simpson’s Rule would have worked out.