Fall 2006
Statistics 702/J702 - Introduction to Statistical Theory I
Tuesday / Thursday 4:00-5:15
203 BA Building

Course Website: http://www.stat.sc.edu/~habing/courses/702F06.html

Assigned Tuesday 8/29 Read for Thursday 8/31
Sections 1.1-1.3 and 1.5-1.6
(not "Law of Total Probability" (pg.18) to end of section 1.5 (pg. 23))
#1 - Due for Tuesday 9/5
1) A fair coin is tossed twice. Let A={first toss is H}, B={second toss is H}, C={first toss=second toss}. Which of A, B, and C are independent of the others? Are A, B, and C mutually independent?
2) Consider the experiment of flipping a fair coin once and rolling a fair six-sided die once. Let the event A={a head and an odd number occurred}.
a) What is the sample space for this experiment?
b) What is the probability associated with each sample point?
c) What sample points make up the event A?
d) Use parts b and c to find P(A).
Assigned Thursday 8/31 Read for Tuesday 9/5
No new reading
#2 - Due for Thursday 9/7
Construct a tree diagram for the example on slide 5 of the class on 8/31. Indicate the probabilities for each branch of the tree, and also identify them symbolically in terms of events A=made first shot, and B=made second shot (e.g. P(A U B), etc...). Use the tree to find the P(made both), P(made exactly one), and P(missed both).
Assigned Tuesday 9/5 Read for Thursday 9/7
Sections 1.5 and 1.4
#3 - Due for Tuesday 9/12
Chapter 1: #12, #36, #53
Chapter 2: #27, but use a population of size 500, not 100,000
Assigned Thursday 9/7 Read for Tuesday 9/12
Make Sure Chapter 1 is OK
#4 - Due for Thursday 9/14
Chapter 1: #17 - You may choose either binomial or hypergeometric for this problem. For your choice, describe the "population", give the formula when p (the percentage of defectives) =0.2 and use R to make the graph, plotting the values for p=0, 0.05, 0.1, 0.15, 0.2, and 0.25.
Assigned Tuesday 9/12 Read for Thursday 9/14
Sections 2.1, 2.1.1, 2.1.2, and 2.1.4
#5 - Due for Tuesday 9/19
Chapter 2: #1 (also calculate the mean and variance), #11 (remember, you can discuss this with each other, ask me for help, and there is partial credit!)
Assigned Thursday 9/14 Read for Tuesday 9/19
None this time
#6 - Due for Thursday 9/21
Chapter 4: #7
Assigned Tuesday 9/19 Read for Thursday 9/21
Sections 2.1.3 and 2.1.4
#7 - Dues for Tuesday 9/26
Wisconsin has approximately 4,000,000 registered voters, of which 4 percent are undecided for the upcoming election.
a) Briefly, why is it not unreasonable to model a survey of 100 of these voters as a binomial rather than a hypergeometric?
b) In a random sample of 100 registered voters, what is the probability of having no undecided respondents?
c) How many do you expect to have to survey before you have the first undecided respondent?
d) What is the probability that the tenth person you talk to is your second undecided?
Assigned Thursday 9/21 Read for Tuesday 9/26
Section 2.1.5
No assignment made
Just try the last two example problems
Assigned Tuesday 9/26 Read for Thursday 9/28 Pg. 47-49 No homework assigned
Exam is due at noon on the 3rd
Assigned Thursday 9/28 Read for Tuesday 10/3
Sections 2.2.1 - 2.2.3
No assignment made
Exam is due at noon the the 3rd
Assigned Tuesday 10/3 Read for Thursday 10/5
Pages 58-59 and example B on page 61.
#8 - Due for Tuesday 10/10
Chapter 2: #40, #45
Assigned Thursday 10/5 Read for Tuesday 10/10
Rest of section 2.3
#9 - Due for Thursday 10/12
Chapter 2: #67. Also use R to plot the pdf for a few values of alpha and beta to demonstrate how they affect the behavior of the Weibull distribution.
Assigned Tuesday 10/24 Read for Thursday 10/26
Sections 3.1-3.6
#10 - Due for Tuesday 10/31
Homework 10 Word File
Assigned Thursday 10/26 Read for Tuesday 10/31
Section 3.7
No assignment made
Make sure you are caught up on the readings.
Assigned Tuesday 11/14 Read for Thursday 11/16
Sections 4.1-4.5
#11 - Due for Tuesday 11/21
Chapter 4: 49 a-b
and
Let X and Y be independent Gamma with parameters lambda and alphaX and alphaY. Use M.G.F.s to find the distribution of X+Y.
Assigned Tuesday 11/21 Read for Tuesday 11/28
None this time
#12 - Due for Thursday 11/30
Chapter 1: #74
Chapter 4: #67
Assigned Thursday 11/30 Read for Tuesday 12/5
Chapter 5
#13 - Due For Thursday 12/7
1) Consider a binomial with n=5 and p=.2. Estimate P(X<=1) using the central limit theorem, both with and without the continuity correction. Compare these results to the actual value.
2) Read the definition of expected value on page 118 and Example G on page 119. You can use simulation to demonstrate that the central limit theorem doesn't apply to the cauchy.

The code:

x<-rep(0,10000)
for (i in 1:10000){x[i]<-mean(rcauchy(1000))}

will calculate the means of 10,000 samples of size 1,000. Verify that this doesn't seem to be normal.
3) Cauchy random variables don't have moment generating functions (they don't have moments!), but do have characteristic functions that behave similarly. In particular E[exp(itX)]=exp(-|t|) for a Cauchy random variable. Use this to show that the average of two Cauchy random variables is Cauchy. [The rules for sums and multiplication by a constant work the same for moment generating functions and characteristic functions.]