Statistics 583 |
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Tuesday, March 25, 2003
I just posted the solution to homework 7, along with tonight's lecture notes, and the R code for plotting curves of the power function of the Z-test. We will have a quiz next week. The quiz will cover Chapter 7. Cheers, David Wednesday, March 12, 2003
I just posted Homework 7. Its due on March 25th. I'll scan the relevant pages from the book later today. The final will be comprehensive. Quiz 2 will be on April 1 and will cover Chapter 7 and Section 8.1. Quiz 3 will be on April 22. At this stage I have no idea what it will cover - it depends on how far we get. Cheers, David Tuesday, March 11, 2003
Here are the various bits of R code I used tonight in class: varianceR.txt (boxplots of biased and unbiased variance estimators) poissonVarUnbiasedR.txt (variance of different estimators of Poisson variance) asympMLER.txt (asymptotic distribution of MLE) Monday, March 10, 2003
Sunday, March 09, 2003
With regard to Homework 6, the "ten numbers" that Example 7.5.2 presents are 0.86, 1.53, 1.57, 1.81, 0.99, 1.09, 1.29,1.78, 1.29, 1.58 with a sample mean of 1.379 and sigmaPrime = 0.3277. Saturday, March 08, 2003
Friday, March 07, 2003
Its the second probability from (7.3.11): Pr( |sigmaHat - sigma| < sigma/5 ) so that's the prob. that sigmaHat is within one-fifth of sigma, i.e., Pr( 0.8 < sigmaHat/sigma < 1.2 ) then square everything and multiply through by n. Does that help at all? D. Thursday, March 06, 2003
Wednesday, March 05, 2003
Sure, even with just one observation, the likelihood is still the likelihood. If you use the factorization theorem to solve the problem, you do not need the hint. However to solve the problem by the "direct" method (i.e., just using the definition of sufficiency) you need the hint. On Question 1, you want the prior mean lifetime to be 4 months. So, since the mean of an exponential(beta) is 1/beta, its E(1/beta) you want equal to 4. OK? Concerning the post at 4:23pm, the factorization theorem pertains to a factorization of the likelihood, so its not obvious to me how you could use the distribution of Y=X1+X2. I'll try to post some summary of the current homework scores soon. Cheers, David Tuesday, March 04, 2003
I'm having problems with my scanner. I'll try to get the lecture notes and the homework problems scanned tomorrow. D. I've posted the first quiz & solution. I also posted the new homework assignment (just three questions). Take a look at the sentence immediately following the proof of Theorem 6.9.1. That's why you need it. D. You can drop the T because it is a function of the x's. It would like saying whats the joint probability of getting 5 heads and two times the number of heads being 10 in 12 tosses of a fair coin. This is just the same as getting 5 heads in 12 tosses. Does that help? D. The example on the top half of 583-37.jpg is for binomial. You could also do it with Bernoulli's but as written its for binomial.For 6.3.14, it won't make any difference whether you use bernoulli or negative binomial since the combinatorial term disappears into the constant of proportionality. Monday, March 03, 2003
Reading ahead... we will cover all of Chapter 7 except for parts of section 7.8. In Chapter 8, we will omit 8.3 and 8.4. We might also omit 8.2 depending on how things go. D. Sunday, March 02, 2003
Concerning FY Yang's post below... Consider two random variables Z and Y where Z is standard normal and Y is chi-square with n degrees of freedom. Then, the book derives the density of a particular function of these two random variables, namely Z / sqrt(Y/n). This, by definition, is the density of the t-distribution. With a little algebra you can see that (Xbar - mu) / (sigmaPrime / n) is of this form except with n-1 degrees of freedom. The neat thing about this is that it does not depend on the true value of sigma. We were then able to use this to get a confidence interval for mu on page 49 of the lecture notes. Example 7.5.3 uses the delta method. Chapter 5 discusses the delta method. Did you cover the delta method last semester? I'll go over Example 7.5.4 in class. So, for this week's homework, you can either hand it to me in class on Tuesday, or you can fax it to me or email it to me anytime before midnight Friday. Fax it to 973-665-1287. Saturday, March 01, 2003
Here's the little piece of R code for sample size determination that we used in class on Tuesday. D. Concerning the question about p.339 "Sampling from a Normal Distribution", yes, the factor involving the x's does get absorbed into the constant of proportionality. This is often a crucial trick when doing these Bayesian manipulations. Is everyone clear what's going on here? For instance, suppose I am making inference about a probability p, p in [0,1], and I tell you that the density of p is proportional to p^2. Now suppose I want to compute Pr(p>0.9). Well, p^2 does not integrate to 1 over [0,1] (In fact, int_0^1 p^2 dp = 1/3) so I seem to be stuck. However, we know that since the density (call it [p]) is proportional to p^2, then [p] = cp^2 for some c. But the density must integrate one so c must be 3. Now I can compute Pr(p>0.9). It is given by: int_0.9^1 3 p^2 dp which is 0.729. You can also get R to provide the answer - just type 1-pbeta(0.9,3,1). I'm reluctant to postpone the homework assignment by a whole week. If I postpone it to Friday, how can you get it to me? Fax? E-mail? Would that work? D. |
