STAT 6160

Statistics

STAT 6160 is the first semester of a two-semester graduate sequence on Mathematical Statistics. This first course is an introduction to the theoretical probability used in statistics, with an emphasis on the mathematical theory. There will be a rigorous treatment of random variables, their probability distributions, and mathematical expectations in a univariate and multivariate setting. Also included are conditional probabilities; stochastic independence; sampling theory; and limit laws.

Textbook information:

Introduction to Mathematical Statistics, 6th Edition
Hogg, McKean, Craig

Pearson Prentice Hall, 2005
ISBN: 0-13-008507-3

A list of errata (typos and errors) can be found here.

NOTE: This is NOT the same text used last time this course was offered.

Course Syllabus for Spring 2008

Date Assigned	Date Due	Homework
1/14/08	1/28/08	p 9, #1.2.3, 1.2.4, 1.2.6, 1.2.8, 1.2.10, 1.2.11, 1.2.12.
1/14/08	1/28/08	Download the R statistical language. http://www.r-project.org/ . Click on the link under "Download" on the left of the screen, choose a mirror site, then follow directions to download and install R. Type the following in R. x <- runif(100, min=0, max=1) hist(x) Use the up arrow to repeat these commands several times. Then repeat these commands, but change the 100 to 1000. Write a few sentences about what is similar and what is different each time. You do not need to show me your graphs.
1/16/08	1/28/08	p 19, #1.3.1, 1.3.4, 1.3.5, 1.3.7, 1.3.11, 1.3.13, 1.3.20, 1.3.21, 1.3.22, 1.3.23
1/23/08	2/4/08	p 29, #1.4.1, 1.4.5, 1.4.8, 1.4.11, 1.4.14, 1.4.15, 1.4.18, 1.4.22
1/23/08	2/4/08	Read about the Monty Hall dilemma. ( p32 number 1.4.30, http://www.math.uah.edu/stat/games/MontyHall.xhtml ) If there were five doors, four of which conceal a goat, one of which conceals a car, and Monty revealed three goats before giving the option to switch, what would the probability of winning be if you switched, and what would the probability of winning be if you did not switch?
1/30/08	2/11/08	p 40, #1.5.1, 1.5.4, 1.5.5, 1.5.6, 1.5.8, 1.5.9
1/30/08	2/11/08	In R, pick 10 uniform random numbers ( x <- runif(10) ) and generate and plot the empirical cumulative distribution function for your numbers ( plot(ecdf(x)) ). Print your graph and determine from it . Compare this to where is a uniform random variable.
1/30/08	2/11/08	p 44, #1.6.2, 1.6.3, 1.6.4, 1.6.7, 1.6.8
2/4/08	2/18/08	p 51 # 1.7.1, 1.7.6, 1.7.8, 1.7.9 a and b, 1.7.10, 1.7.14, 1.7.18, 1.7.20, 1.7.24. For extra credit, look at 1.7.4, 1.7.9 c, and 1.7.22. (Hint: look up the integral and derivative of the arctan function.)
2/6/08	2/18/08	p57 #1.8.2, 1.8.3, 1.8.6, 1.8.8, 1.8.12, 1.8.13.
2/13/08	2/25/08	p64 #1.9.1, 1.9.2, 1.9.3, 1.9.4, 1.9.6, 1.9.8, 1.9.17
2/13/08	2/25/08	Calculate the mean and variance of a uniform distribution on the interval (0,1). Then, in R, generate a random sample of 100 uniform(0,1) random variables (x <- runif(100,0,1) ). Calculate the mean and variance of the sample (mean(x) and var(x) ). Repeat this sequence of commands several times and comment.
2/25/08	2/27/08	The midterm exam will be handed out on Monday the 25th and will be due at the beginning of class the 27th.
2/18/08	2/25/08	p72 #1.10.1, 1.10.2, 1.10.3, ~~1.10.6~~
2/18/08	2/25/08	· Let D be a collection of subsets of the space C. Let B be the sigma-field generated by D. Let and be probability measures on C that coincide on D, that is for all D in D. Show that and must coincide on B also. · Do problem 1.4.9 on page 30. · Prove that if is a random variable defined on the interval and if exists, then . · Let be a continuous random variable with cumulative distribution function. Let be defined by . Show that is uniformly distributed on the unit interval. · Let be a uniformly distributed random variable on the unit interval, and let be a continuous cumulative distribution function. Let be defined by . Show that the cumulative distribution function of is . · Suppose and are random variables and that for all x. Which is true? Why? o o o We cannot tell which expected value is larger.
2/27/08	3/10/08	p 82. #2.1.1, 2.1.6, 2.1.8, 2.1.10, 2.1.13, 2.1.14
2/27/08	3/17/08	p 92. #2.2.2, 2.2.3, 2.2.5, 2.2.6
2/27/08	3/17/08	Let X and Y be independent uniform(0,1) random numbers. Find the pdf of X+Y and graph this function. In R, pick 1000 uniform random numbers X ( x <- runif(1000) ) and 1000 uniform random numbers Y ( y <- runif(1000) ). Then calculate the vector of sums Z (z <- x+y). Compare the first elements of X, Y, and Z (x[1], y[1], z[1]) to check that Z is the sum of the values in X and Y. The create a histogram of the values in Z (hist(z)) and compare it to the graph you made.
3/10/08	3/17/08	p 99. #2.3.1, 2.3.2, 2.3.3, 2.3.8, 2.3.9, 2.3.12
3/12/08	3/24/08	p107 #2.4.1, 2.4.3, 2.4.6
3/17/08	3/24/08	p114 #2.5.1, 2.5.2, 2.5.3, 2.5.8, 2.5.9
3/17/08	3/24/08	p122 #2.6.1, 2.6.3, 2.6.4, 2.6.8
3/24/08	3/31/08	p 140 #3.1.1, 3.1.3, 3.1.4, 3.1.6, 3.1.8, 3.1.12, 3.1.14, 3.1.21
3/24/08	3/31/08	p 147 #3.2.2, 3.2.7, 3.2.8, 3.2.10
3/31/08	4/7/08	p 157 #3.3.3, 3.3.4, 3.3.6, 3.3.10, 3.3.11, 3.3.15, 3.3.18, 3.3.25 a
3/31/08	4/7/08	p 168 #3.4.2, 3.4.3, 3.4.7, 3.4.10, 3.4.16, 3.4.19, 3.4.20, 3.4.28
3/31/08	4/7/08	In R, the command pnorm(10, mean=5, sd=3) will give the probability that X is less than or equal to 10, if X is a normal random variable with mean 5 and standard deviation 3. Use similar commands to find: P(X > 100) if X is normal with mean 110 and standard deviation 20. P(Y < 25) if Y is gamma with shape parameter 5 and scale parameter 20 P(X > 10) if X is Poisson with mean 10.
4/7/08	4/14/08	p 188 #3.6.2, 3.6.4, 3.6.8, 3.6.9, 3.6.10, 3.6.12
4/14/08	4/21/08	p 201 #4.1.2, 4.1.4, 4.1.8, 4.1.9, 4.1.11, 4.1.22, 4.1.26
4/14/08	4/21/08	p 207 #4.2.2, 4.2.3, 4.2.4, 4.2.5
		p 218 #4.3.1, 4.3.2, 4.3.5, 4.3.7, 4.3.11, 4.3.16
		p 225 #4.4.1, 4.4.8, 4.4.9, 4.4.10
		The following set of commands generates 2000 random uniform numbers, rearranges them into 2 rows and 1000 columns, then plots the histogram of the column means (2 uniform numbers per mean). > x <- runif(2000) > dim(x) <- c(2,1000) > hist(colMeans(x)) Use similar commands to generate exponential random variables with rate 2, and display histograms of sample means for n=2, 5, 10, 50. Comment. Then do the same, but using Cauchy random variables instead of exponential. Comment again. Why is there a difference in behavior here?