Exercises for STK4011/9011 autumn 2015

Below will be given exercises for the coming week(s) and an overview of exercises that have been given to earlier weeks. Unless otherwise stated, exercises are from the book by Casella and Berger.

Exercises to coming week(s)

Exercises to earlier weeks

Week 49 (3 December):

This is the last week of teaching, and we will consider the following exercises:.

• Exam autumn 2013: Problems 1 and 2.

Week 48 (Solutions):

• Do the following exercises from chapter 9: 9.4 and 9.6.
• Additional exercise: Let \(X_1, \ldots ,X_n\) be independent and Poisson distributed with parameter \(\lambda\). Derive approximate \(1-\alpha\) confidence intervals for  \(\lambda\) by inverting a) the Wald test, b) the score test, and c) the likelihood ratio test.
• Exam autumn 2014: Problems 1 and 2.

Week 47 (Solutions):

• Do the following exercises from chapter 8: 8.20, 8.22ac, 8.25bc, 8.27, 8.31 and 8.34b.
• Do the following exercise from chapter 10: 10.34a (question b is optional).
• Additional exercise: Look at slide 21 from the lectures in week 46. Derive an expression for \(-2\log\lambda({\bf X})\).

• Do the following exercises from chapter 8: 8.3, 8.6, 8.15 and 8.17.

Week 45 (Solutions):

• Do the following exercises from chapter 10: 10.1, 10.3, 10.5, 10.6 and 10.9.

• Do the following exercises from chapter 5: 5.33, 5.35, 5.41, 5.42 and 5.44.

• Do the following exercises from chapter 7: 7.47, 7.52, 7.59 and 7.60.

• Do the following exercises from chapter 7: 7.19, 7.38, 7.41 and 7.42. (By a mistake you got a copy of the solution of 7.19 with the solutions to week 40. Try to do the exercise without looking at this solution.)

• Do the following exercises from chapter 7: 7.9, 7.11, 7.13, 7.22, 7.24 and 7.40. (In order to find the mean and the variance of the MLE in exercise 7.11, you should note that the MLE is of the form n/T, where T is gamma-distributed with shape parameter n and scale parameter 1/theta, and then use the result of exercise 3.17.)

• Do the following exercises from chapter 6: 6.2, 6.3, 6.4, 6.5, 6.6, 6.8 and 6.9a-c.

Week 38 (Solutions)​​

• Do the following exercises from chapter 5: 5.5, 5.6, 5.24, 5.25 and 5.26. (Before you do exercise 5.26, you should study carefully the proof of Theorem 5.4.4 on page 229.)
• Read section 5.3.2 (pages 222-225) and do exercises 5.17 and 5.18a-b. [In 5.17a you should start with U and V independent and chi-squared distributed variables with p and q degrees of freedom, and consider the transformation X=(U/p)/(V/q) and Y=U+V. In 5.17b you may use the result in exercise 3.17 (which is valid for v>-alpha) and the fact that the chi-squared distribution is a special case of the gamma distribution.]

• Do the following exercises from chapter 4: 4.4a-b, 4.5, 4.21, 4.22 and 4.27.
• Additional exercise: Look at slide 24 from the lectures in week 36. a) Prove the result on moment generating functions stated on the first half of the slide. b) Prove the result on the distribution of linear combinations of independent normal random variables given at the bottom half of the slide.

• Postponed exercises from week 35: 3.13, 3.14, 3.17, 3.38 and 3.39.
• Exercises on exponential family of distributions: 3.28b-e (consider only the case where both parameters are unknown in b and c), 3.29, 3.30b, 3.32 (use the result on slide 7 for the lectures in week 35), 3.33a-c. In exercise 3.30b one should replace "beta(a,b)" by "Poisson(\(\lambda\))", see the errata to Casella and Berger.

• ​​​Read section 2.1 and do exercises 2.1 and 2.6.
• Read pages 62-68 in section 2.3 and do exercises 2.30 and 2.33.
• Do the following exercises from chapter 3: 3.13, 3.14, 3.17, 3.38 and 3.39 (postponed to week 36).