P609 Homework Assignment #5
Due Tuesday, April 12, 2016
Problem 1
The local newspaper once carried a story about jury selection for a trial.
The paper stated:
At a hearing Thursday in Owen County, Nardi denied a motion filed by
defense attorneys David Colman and Elizabeth Cure to disqualify the
jury panel. Colman and Cure observed that out of 80 potential jurors,
only one was younger than 30.
"Monroe County has a population of 120,000 people, of which between
30,000 and 40,000 are students at Indiana University," the motion stated.
"A truly random jury-selection process for Monroe County should result
in no fewer than 20 students from Indiana University."
This might be a good time for a lawyer joke, but I will resist that temptation
and ask you to analyze this situtation.
a) Assuming that there are 120,000 people in Monroe county and 30,000 of
those people are students at IU, what is the probability that a group of
80 randomly selected people would include fewer than 20 students?
Note, if you had been among the people called for jury duty, you probably would
have been excused from jury duty if you had pointed out the lawyer how stupid
his statement was. (I got out of jury duty a couple of years ago when I told
the laywer that he had just contradicted himself.)
b) What is the probability if 40,000 of the 120,000 people, that there would be
fewer than 20 students in a randomly selected group of 80 people?
Note: I will be pleased if you can use Mathematica to solve this problem, but
if you prefer another approach, that is also fine.
Problem 2
Complete Problem 11.13 parts (c-e) on page 432 of CSM.
Problem 3
The Gaussian Distribution and the Metropolis Method (See section 11.7
starting on page 435 of CSM and particularly problem 11.17 on page 437):
a) Write a program to generate a Gaussian distribution using the
Metropolis algorithm as described in class and in Sec. 11.7 of CSM.
However, instead of using a value \delta_i for the trial step
uniformly distributed in the interval [-\delta, \delta], I would
like you to use a triangular distribution in the interval
[-\delta, \delta]. In class we discussed how to do this by
using two uniformly distributed random numbers r_1 and r_2.
Alternatively, you might consider the inverse transform method.
b) Try multiple values of delta and create runs with 50000 numbers.
Plot histograms of the entire runs and time histories of the first
5000 values.
c) Plot the acceptance ratio as a function of delta.
d) Using the program autocorr (Usage: autocorr maximum_lag