NTHU MATH 2820 - Statistics
清華大學 數學系 統計學
Feb ~ Jun 2017
Notes
(Jun 16) 第六套投影片(Decision Theory and Bayesian Inference)在學期中未教完的部份，將會於七月錄製影音檔，放置於課程網站上。有興趣的同學可於 暑假時撥空點閱學習。 (Jun 16) 作業總成績, 期末考成績, 學期總成績, 及成績統計 (Jun 16) 期末考解答 (Jun 08) Extra office hour before final exam: time, 6/12 (monday) 1~5 PM; location: 綜合三館 room 818. (Jun 01) 期末考考古題及解答。 (Jun 01) 期末考資訊及注意事項。 (May 25) 端午節放假補課方式：將於5/28（周日）7:00PM前，於課程網站上，放置兩小時影音檔，以供同學們點閱學習。 (May 25) 下周二上課日(5/30)，適逢 端午節放假，停課一次。 (May 23) 廣告: 中央研究院統計科學所將於9/6-9/7為大學生舉辦2017統計科學營,歡迎各位報名參加。 (May 08) 期中考解答及成績統計。 (Apr 19) 期中考考古題：考題1(解答1)，考題2(解答2)。 (Apr 19) 期中考資訊及注意事項。 (Apr 04) 廣告: 中央研究院統計科學所將於7/5-7/21為大學生舉辦2017統計研習營,歡迎各位報名參加。 (Mar 30) 清明節放假補課方式：將於4/2（周日）7:00PM前，於課程網站上，放置兩小時影音檔，以供同學們點閱學習。 (Mar 30) 下周二上課日(4/4)，適逢清明節放假，停課一次。 (Feb 23) 228放假補課方式：將於2/25（周六），於課程網站上，放置兩小時影音檔，以供同學們點閱學習。 (Feb 23) 下周二上課日(2/28)，適逢228和平紀念日放假，停課一次。 (Feb 15) 有關助教及其office hour的資訊，請見Syllabus
Lecture (You need to download and install Adobe Reader for reading the lecture notes.)
Lecture Notes
Lecture Notes with Hand-Written Notices
Video 01 (video 1) (video 2) 02 (video) Feb 21 (video 1) (video 2) Feb 23 (video) Feb 28 (228放假補課) (video 1) (video 2) Mar 02 (video) Mar 07 (video 1) (video 2) Mar 09 (video) Mar 14 (video) 03 Mar 14 (video) Mar 16 (video) Mar 21 (video 1) (video 2) Mar 23 (video) Mar 28 (video 1) (video 2) Mar 30 (video) Apr 04 (清明節放假補課) (video 1) (video 2) Apr 06 (video) Apr 11 (video 1) (video 2) Apr 13 (video) Apr 18 (update) (video) 04 Apr 18 (video) Apr 20 (video) Apr 25 (video) 05 Apr 25 (video) Apr 27 (video) May 04 (video) May 09 (video 1) (video 2) May 11 (video) May 16 (video 1, missing) (video 2) May 18 (video) May 23 (video 1) (video 2) May 25 (video) 06 Decision Theory and Bayesian Inference (part 1) (part 2) (part 3)
May 30 (端午節放假補課) (video 1) (video 2) Jun 01 (video) Jun 06 (video 1) (video 2) Jun 08 (video)
Assignment and solution
Homework Questions Due Day Solution 1 Ch 1. #54, #59, #71, #74, #78
Ch 2. #6, #33, #40
Ch 3. #1, #7, #8, #10
Mar 02 sol 2 Ch 2. #66, #67
Ch 3. #16, #21, #25 [Hint: Let W=±1 with probability 1/2 each and W independent of X, then Y and WX have same distribution], #53 [Hint: Here is one way to get the answer. Let Y_{1}=U_{1}U_{2}/(U_{3}^{2}), Y_{2}=U_{1}, Y_{3}=U_{2}. Then, you can use Theorem2.7 in LNp.32 to get the joint pdf of Y_{1}, Y_{2}, and Y_{3}. Note that the joint pdf is zero outside the region 0<Y_{2}<1, 0<Y_{3}<1, Y_{2}Y_{3}<Y_{1}<∞. Use the joint pdf to obtain P(Y_{1}<1).], #64, #66, #70, #77
Ch 4. #6, #13 [Hint: Note that you can replace x by ∫_{0}^{x} 1 dt ]
Mar 09 sol (revised) 3 Ch 4. #19 [Hint: The expectation can be obtained by only using the marginal distributions of U_{(k)} and U_{(k}_{-}_{1)}], #26, #42 [Hint: Use the cdf, mean, variance equations given in LNp.71 to find the probability], #49, #54, #60, #61, #67, #75 [Hint: Use the means and variances given in LNp.70 and p.71. Apply law of total expectation and variance decomposition to find the mean and the variance of U, respectively.], #76, #79, #100 [Hint. Apply δ-method to find the approximate mean and variance.]
Mar 16 sol 4 (turn-in questions)
Ch 2. #21 (Note. This is called memoryless property) [Hint: check LNp.62 for the cdf of geometric distribution], #31, #48 [Hint: check LNp.71 for the cdf of exponential], #61 [Hint: You can show it by using mgf].
Ch 3. #22 [Hint: You can first show that: If X_{1}~P(λ_{1}), X_{2}~P(λ_{2}), and X_{1}, X_{2} are independent, then the conditional distribution of X_{1} given that X_{1}+X_{2}=n is binomial distribution B(n, λ_{1}/(λ_{1}+λ_{2})).], #26 [Hint: use beta function in LNp.75], #47.
Ch 4. #30 [Hint: try sum-to-one method], #79, #80, #91, #92 [Hint: use the law of total expectation to find the mgf of X (check LNp.71 and LNp.73 for the mgf of exponential and gamma, respectively), and compare it with the mgf of negative binomial distribution given in LNp.63].
(no-need-to-turn-in question)
Please practice the exercises (i.e., those marked as "Ec") given in Lecture Notes with Hand-Written Notices, Ch 1-6, p.58-83, as many as possible.
Mar 23 sol 5 Ch 5. #1 [Hint: use Chebyshev's inequality and imitate the proof in LNp.93], #2 [Hint:
], #4, #5 [Hint: use the theorem: if a_{n}→a, then (1+a_{n}/n)^{n}→e^{a}. The mgfs of binomial and Poisson are given in LNp.60 and LNp.67, respectively.], #11, #13 [Hint: use normal approximation to binomial], #16 , #17 [Hint: use CLT], #21 [Hint: For (c), by Cauchy-Schwarz inequality, we have
], #28.
Ch 6. #8 [Hint: You can use mgf to relate the distributions of 2X and 2Y to the chi-square distribution], #11.
Mar 30 sol 6 Ch 8. #7(a)(b) [Hint: the 1st moment of geometric distribution is given in LN, ch1-6, p.62], #9 [Hint: check the diagram in LNp.12], #16(a)(b) [Hint: The 1st moment is zero, and you can use gamma pdf and gamma function given in LN, ch1-6, p.73&74, to find the 2nd moment], #18(a)(b), #21(a)(b) [Hint: For (a), let Y=X-θ, then Y~exponential(1), i.e., 1=E(Y)=E(X-θ). For (b), note that x_{i }≥ θ for i=1,...,n, i.e., x_{(1) }≥ θ.], #26 [Hint: Try to use the hypergeometric distribution given in LN, ch1-6, p.68, to model the data.], #51 [Hint: In this case, median is the (m+1)^{th} smallest observation, i.e., X_{(m+1)}. You can first show that if X_{(m+2)}≥θ≥X_{(m)} , then Σ|X_{i}-θ|=|X_{(m+1)}-θ|+(X_{(m+2)}-X_{(m)})+(X_{(m+3)}-X_{(m}_{-1)})+...+(X_{(2m+1)}-X_{(1)}). Then, try to generalize the result to obtain a general statement.], #60(a)(b)(c)(d) [Hint: For (b) and (d), use the results given in LN, ch1-6, p.73&74].
Apr 6 sol 7 Ch 8. #7(c), also, obtain the Fisher information of X and identify the asymptotic sampling distribution of the MLE [Hint: the mean of geometric distribution is given in LN, ch1-6, p.62], #16(c), also, obtain the Fisher information of the i.i.d. sample and identify the asymptotic sampling distribution of the MLE [Hint: you can use gamma pdf and gamma function given in LN, ch1-6, p.73&74, to find E|X_{i}| or E(X_{i}^{2}).], #18(c), also, obtain the Fisher information of the i.i.d. sample and identify the asymptotic sampling distribution of the MLE, #53(a)(b)(d) [Hint: (i) the MLE and moments estimator are given in LN with Hand-Written Notices, Ch8, p.22. (ii) For (d), find an estimator which is a function of the MLE and is unbiased], #58(a)(b), also, identify the asymptotic sampling distribution of the MLE [Hint: the log-likelihood function is given in LNp.25 and notice that the marginal distributions are X_{1}~B(n, (1-θ)^{2}), X_{2}~B(n, 2θ(1-θ)), X_{3}~B(n, θ^{2}).
Apr 18 sol 8 Ch 8. #16(d), also, show that the pdfs form an exponential family and find a sufficient and complete statistic; #18(d), also show that the pdfs form an exponential family and find a sufficient and complete statistic; #21(c), also show that X_{(1) } is complete by definition and examine whether the pdfs form an exponential family; #49; #53(c); #57; #60(e), also find the Cramer-Rao lower bound and show the MLE achieves the lower bound; #72, also show that the gamma distribution form a 2-parameter exponential family and show that ΠX_{i} and ΣX_{i} are sufficient and complete.
Apr 25 sol 9 (no-need-to-turn-in)
sol (revised) 10 May 09 sol 11 Ch 9. #1, #2, #3, #5, also if false, please explain why.
May 16 sol 12 May 25 sol 13 Ch 9. #12, #13(a)(b)(c), #24
,
#26, also if false, please explain why. #37 [Hint: you may model the numbers of deaths as random variables with independent Poisson P(λ_{i}), i=1,..., 12, and examine whether λ_{i}'s are equal], #40 [Hint: p_{1}+p_{2}=1 and X_{1}+X_{2}=n], #43 [Hint: you can use GLR test for goodness-of-fit or Pearson's Chi-square test, and compare your answer with the solution given in textbook, page A40], #44 [Hint: Let O_{1}, O_{2}, and O_{3} be the numbers of AA, Aa, and aa, respectively. Then, (O_{1}, O_{2}, O_{3})~Multinomial(n, (1-θ)^{2}, 2θ(1-θ), θ^{2}). The MLE of θ, where 0<θ<1, is (2X_{3}+X_{2})/(2n) (see LN, CH8, p.25). Use O_{1}, O_{2}, O_{3} to derive the likelihood ratio test statistic. By the way, you may compare this question and the Example 7.19 given in LN, CH9, p.43. Try to find the differences on their H_{0} and H_{A} (or the differences on their Ω_{0} and Ω).].
Jun 06 sol (revised) 14 (no-need-to-turn-in)
textbook (2nd ed.) Ch 15. (Problem statements) #1, #6, #13, #15, #23, #24, #29. (blue-colored problems are not included in final exam)
sol