Homework 1.  Statistics 771, Spring 09
Posted online Thursday January 21/09
Due in class Monday Feb 2.


1.	The Gamma(shape=a,rate=1) distribution has probability density 

	f(x;a,b) = c x^{a-1} exp( - x ) 

	for x >= 0, where c = 1/Gamma(a) is a normalizing constant,
	and a > 0 .  The cumulative distribution function is

	F(x;a) = \int_0^x f(y;a)  dy

	Note that Gamma(a+1) = a Gamma(a) and integration by parts to
	establish

	**	F(x;a) = F(x,a+1) + exp(-x) x^a / Gamma(a+1)

	Use ** to establish further that

	F(x;a) = exp(-x) sum_{n=0}^\infty [  x^{a+n} ]/[ Gamma(a+n+1) ]

	Describe a numerical approximation to F(x;a).  How would you compute
	terms involving the Gamma() function?

	Implement your algorithm for all pairs of  a = 1, 2, 4, 8, 16 and
	x = 1,2,4,8, 16, and compare to the R function pgamma().


2.	The file  

	http://www.stat.wisc.edu/courses/st771-newton/hw/rcc-distribute.txt 

	holds an R data object (a list) with 
	results of a study (by Jiang and Moch) of genomic damage in a sample 
	of n=116 renal cell cancer tumors. Damage was measured using comparative
	genomic hybridization (CGH), and records whether or not amplifications
	or deletions occur in various genomic regions.  (The data file
	has results for m=72 possible aberrations in each tumor.)

	The data may be loaded into R and examined via

		rcc <- dget("rcc-distribute.txt")
		x <- rcc$data
		names(rcc)   ## shows components of the list

	The problem is to write some R code (ideally a function) that
	takes a matrix like x and returns a visualization of x in which
	the matrix is layed out and plotted in two colors and in which row sums 	and column sums are arranged in barplots on two sides of the image.

3.	Read about the QR decomposition at
	http://en.wikipedia.org/wiki/QR_decomposition
	and read chapters 1 and 2 of Givens and Hoeting.