October 13, 2013 --- Class 12 --- Random Walks and Biased Estimators
Using Mathematica
Activities:
Random Walks Using Mathematica
Working through section G of ~sg/mathematica_notes_sg_extended.pdf
we learned about several functions in Mathematica that can be
used to create random walks on the integers. We used ListPlot to plot
the walk. We used several functions of use in statistics such a
Histogram, Mean, Variance.
Biased Esimators Using Mathematica
We considered the problem posed in section H of the handout.
Suppose we have a distribution of velocities and we want to
know the travel time for an entity traveling with the average
velocity. The natural thing to do is to measure some velocities
and estimate the average from the measurements. Dividing the
distance by the average, we estimate the time for an entity moving
with the average velocity. However, this is a biased estimate.
A very simple example involves a uniform distribution between zero
and one. The average is 1/2. If the distance is 1, the time for
average velocity is 2. However, with a small sample of measurements
we do not get 2, even as the average of many repetitions of the
experiment.
We started by teaching Mathematica to do multiple integrals
analytically and then having it get a numerical result as described
in the handout.
Several Mathematica notebooks contain examples of the commands
used: ~sg/Documents/feb16_v1.nb , ~sg/Documents/feb16_v2.nb and
~sg/Documents/October_10_2006.nb .
We also looked at the on-line help features, including the
Function Naviator.
In our next class we will consider how to reduce bias.