This project is not yet available in postscript format. It contains many hand drawn diagrams. It will be available as soon as I am able to scan it in. Here, however, is a summary of the project.
Density and Mass
I give a defn of the density as a derivative of the mass of a rod. The first problem is an estimate of density given some information about a given rod. I then explain how to find mass as an integral of the density and explain it in terms of Riemann sums. This includes diagrams. I follow with a problem about a rod with an exponentially decaying density in kg/m.
I discuss the atoms in the rod and have a problem in which the students must compute the number of atoms in various sections of a given rod.
At this point I define the probability that a given atom is located in a given section of the rod. Here the students must divide the number of atoms in that section of the rod by the total number of atoms. The students must perform integration to find these numbers. I ask them to explain what the probability = 1 must mean.
Probability Density
I introduct probability density by comparing it to mass density. I have the student compute some probability densities by taking limits of probabilities divided by smaller and smaller lengths.
I then explain how to find the probability given the probability density and have the students write the appropriate Riemann sum. This is followed by a simple problem.
Lifetimes of Lightbulbs
Here I introduce the lifetime of a lightbulb and discuss its probability density. I also define the "life expectancy" of the lightbulb. I follow this with a few problems which thoroughly analyze the lifetime of a lightbulb, including:
Compute the probability of a lightbulbs failure before 100 hours
Given 50 lightbulbs, estimate how many will fail before 100 hours.
What is the chance that a lightbulb will last more than 200 hours? If you have 570 lightbulbs, approximately how many will last more than 200 hours?
I include a graph of the probability density and ask the students to explain what the areas under various parts of the graph correspond to.
I have the students estimate probabilities by examining the graph.
I have the students draw their own graphs.
Expected Value
Now I introduce the "expected lifetime" or the avergae lifetime of a lightbulb. I give a chart of the lifetimes of a set of lightbulbs and have the students compute the average by adding it up.
I then ask the students to graph the chart with the lifetime on the xaxis and the number of lightbulbs on the y axis. On the same graph, I have them draw the graph of a specific exponentially decaying function. The students then compare this function to their graph and compute various quantities using it. There is a discussion of how the this probability density function corresponds to that set of lightbulbs.
The formula for the expected lifetime is now provided as an integral weighted by the probability density and compared to its Riemann sum.
The students compute the expected lifetime using the probability density function and compare it to the average they computed earlier.
The students now use the formula to find the expected location of an atom in a rod using the probability density from the first section of the project. It is explained that this is the center of mass.
Bell Curves
I explain what the bell curve or normal distribution is and provide an example of such a function. I explain that it is used to study standardized tests.
I have the students verify that the integral of the example is almost 1 by using Simpson's Rule on a sufficiently large interval. I ask them to find an upper bound on the error involved in this approximation. These techniques were taught in the class's text.
The students then integrate to find the expected value. No approximation is necessary here.
The students find the expected value of a subset of the entire population. In particular, I ask them to find the average grade of the people who score above a certain grade on a standardized exam.
The project ends with a sheet containing all the formulas taught inside inside the project.