#################
# QERM 598 	#
# Homework 1	#
# Due 1.17.2007 #
#################

# 1.  Look up ?qnorm, ?pnorm, ?dnorm and ?rnorm in the help files.
# Let Z~N(0,1) and X~N(3,2) 2 = variance, not st.dev.
# Use R to calculate
# a. Pr(Z<.4)
# b. Pr(X>2)
# c. Pr(-1<Z<1)
# d. Solve for a: Pr(Z<a)=.6
# e. Solve for a: Pr(-a<X<a)=.5
 

# 2. The t-distribution.
#
# When X~N(mu,sigma^2), we know (X.bar-mu)/(sigma/sqrt(n)) ~ N(0,1).
# However, using this fact to conduct tests about mu require knowing
# sigma, which is rarely the case.  When we want to know something
# about mu but need to use sigma for inference, we call sigma a 
# 'nuisance parameter', i.e. not the true focus of interest, but
# necessary to proceed.  Estimation in the presence of nuisance parameters
# is a big topic in statistics and much work has been done on it.
#
# One of the first big developments on this process was by Gossett, who
# worked for Guinness and was forced to publish pseudonymously as "Student."
# He worked out the distribution of (X.bar-mu)/(s/sqrt(n)), where
# s is the sample standard deviation.  This distribution is called "Student's t"
# or just the t-distribution, and it has one parameter, nu = n-1, the degrees
# of freedom (dof).  As the dof increase, i.e as n increases
# and s becomes very close to sigma, the t-distribution converges to the
# standard normal distribution.  This exercise should teach you a little
# about the t distribution.
#
# Pseudo-code is given below in the form of comments.  You supply
# the actual R code and run it to generate the desired results.
# You will want to use ?rt to get started with the t distribution.
#
# a. Convergence to the normal distribution
#
# Plot the density of a t-distribution with 1 dof for value of x between
# -5 and 5 as a black line.  Overlay the densities for t-distributions
# with 2, 5, 10 and 30 dof in different colors.  Finally, overlay the 
# standard normal density as a dotted line.  Experiment with the legend()
# function to see if you can label your lines accordingly.
# What are the qualitative differences between the t-distribution densities
# and the normal density?  Why is this the case? 
#
# You may find the following code useful as an example that you can
# modify...
 x.support<-seq(0,12,length=100)
 x.gamma<-dgamma(x.support,shape=2,scale=3)
 # Gamma(alpha=2,beta=3) i.e. with EX = alpha*beta = 6
 plot(x.support,x.gamma,type="line",lwd=2,lty=2,
 main="Gamma(2,3) density",xlab="x",ylab="f(x)")
#
#
# b. Use of t-distribution when variance is estimated.
# 
# Generate 5000 realizations of size n=8 of a N(4,5) random variable and
# store the standardized sample mean of each realization, i.e. subtract
# the true mean from the sample mean, then divide by the sample
# standard deviation over sqrt(8), and then store the sample mean of these new 
# standardized values.  That is you are storing (X.bar-mu)/(s/sqrt(n)) for
# each realization of length 8.
# 
# Create a histogram of these 5000 standardized sample means.  
# Overlay a standard normal density in blue on the histogram.
# Overlay a t-distribution density with 7 dof in red.  Which one seems to
# capture the histograms pattern better?  What are the qualitative differences
# between the blue and red curves?  What does this say about the distribution
# of the standardized sample mean when the sample variance is used, as opposed to
# the true variance?  
#
# Repeat the above experiment with n=20.  How do the results differ from 
# when n=8?  Did you write your code in such a way that changing the sample
# size is easy?  Repeat with n=100.  How close are the standard normal
# and t-distributions in this case?


# 3. Look up an old homework problem of your choice from STAT 512.
# Use R to investigate what's going on in this problem and to 
# better understand the solution you found.  (I recommend one of 
# those problems where you have to find the density of some function
# of a random variable, or the min or max...)  You should each pick
# a different problem and come prepared to present your results in
# 3-5 minutes next week.  

# 4. Read the Wikipedia pages for "Hypothesis testing", "Karl Popper",
# "R.A. Fisher."  Some of these pages, especially the one for Popper,
# might be very esoteric; just read for basic ideas and flavor, but
# pay close attention to the section "Philosophy of Science" and
# "the logical asymmetry between verification and falsification"
# in that article.  Try to figure out why I'm having you read this.

# Suggested background:  The following website:
# http://www.creem.st-and.ac.uk/ken/mt1007.html
# is for an introductory course taught by Dr. Ken Newman at
# Centre for Research into Ecological and Environmental Modelling
# and School of Mathematics and Statistics
# University of St. Andrews
#
# I've emailed him asking permission to point you to these files, but
# as yet haven't received a response.  Anyway, there are several micro-labs"
# that have some basic R exercises.  You might want to begin by trying some
# of these if the homework has you stumped.