NTHU MATH 2820 - Statistics (undergraduate level)

清華大學 學系 統計學 (大學部課程)

Feb 2020 ~ Jun 2020


Notes

 (Jun 27) 作業總成績、期末考成績、學期總成績,已公布於本課程之iLMS網站。成績之分布,請見成績統計
 (Jun 27) 期末考解答
 (Jun 09) 期末考考古題及其解答
 (Jun 09) 期末考資訊及注意事項
 (Jun 03) [廣告] 中央研究院統計科學所將於9/3-9/4為大學生舉辦2020統計科學營,歡迎各位報名參加
 (May 24) 期中考解答成績統計。期中考成績已公布於本課程之iLMS網站。欲取回考卷者, 請親自至授課老師辦公室(綜合三館818室)領取。
 (May 19) 因新冠肺炎疫情趨緩,將自六月起(第一堂課為6/2, 周二),恢復於實體教室授課。影音檔將於上完課後,才會放置於課程網站上。作業繳交方式則將維持原樣,可繳交紙本或上傳電子檔。
 (May 04) 期中考考古題:考題1解答1考題2解答2),劃紅色刪除線的題目,屬於期中考不考的範圍
 (May 04) 期中考資訊及注意事項
 (Apr 27) 有關助教及其office hour的資訊,略有更改,請見Syllabus (紅字部分)。
 (Apr 22) [補課結束] 自下周四(4/30)起,補課結束。每周四的上課時間,調回原本的一小時(11:00AM-12:00PM)
 (Mar 30) 本周四上課日(4/2),適逢民族掃墓節暨兒童節補假,停課一次。
 (Mar 25) 為了因應清大疫情狀況,自即日起,本課程將暫時改為非同步遠距教學。遠距教學之施行方法如下
 (Mar 03) 有關助教及其office hour的資訊,請見Syllabus
 (Mar 03) [補課資訊] 在期中考(時間未定)之前,每周四12:00-1:00PM將在綜合三館203室(上課教室)補課一小時。期末考後則不再上課。補課之影音檔,課後即會放置於課程網站。

 

Lecture Notes

Lecture Notes with Hand-Written Notices

Video

01

 Introduction - what is statistics?

Mar 03


(3394 views)

(1460 views)
02

 Probability

Mar 05


(1779 views)

(1101 views)
Mar 10

(1019 views)

(664 views)
Mar 12

(819 views)

(658 views)
Mar 17

(735 views)

(636 views)
Mar 19

(800 views)

(725 views)
Mar 24

(900 views)

(518 views)
03

 Point Estimation

Mar 24

(1177 views)
Mar 26

(1011 views)

(756 views)
Mar 31

(739 views)

(710 views)
Apr 07

(771 views)

(627 views)
Apr 09

(618 views)

(502 views)
Apr 14

(671 views)

(576 views)
Apr 16

(561 views)

(461 views)
Apr 21

(513 views)

(465 views)
04

 Interval Estimation

Apr 23

(605 views)

(470 views)
Apr 28

(460 views)
05

 Hypotheses Testing

Apr 28

(784 views)
Apr 30

(678 views)
May 05

(694 views)

(605 views)
May 07

(592 views)
May 12

(614 views)

(517 views)
May 14

(551 views)
May 21

(456 views)
May 26

(420 views)

(360 views)
May 28

(362 views)
06

 Decision Theory and Bayesian Inference

Jun 02

(562 views)

(377 views)
Jun 04

(431 views)
Jun 09

(394 views)

(315 views)
Jun 11

(335 views)
Jun 16

(394 views)

(405 views)
Homework Question Due Day Solution Grader
1

Ch 1. #2, #59, #65, #68, #78(a)(b)

Ch 2. #33, #40

Ch 3. #7(also examine whether X, Y are independent random variables and explain why), #8, #21, #25.

Mar 19 sol 沈子翔, 曹詠勛
2

Ch 3. #44, #64 [Hint: the pdf of exponential distribution with parameter λ is λexp(-λx), for x≥0 and 0, otherwise.], #70, #77,

Ch 4. #13, #36, #49, #50, #54, #60.

Mar 26 sol 珮雅, 劉必翔
3

Ch 2. #21 (Note. This is called memoryless property), #30 [Hint: First of all, try to find k0 such that p(k)/p(k+1)1 for any kk0 and p(k)/p(k+1)<1 for any k<k0, where p(·) is the pmf of Poisson.]

Ch 3. #22 [Hint: You can first show that: If X1~Poisson1), X2~Poisson(λ2), and X1, X2 are independent, then the conditional distribution of X1 given that X1+X2=n is binomial distribution B(n, λ1/(λ1+λ2)).], #26 [Hint: use beta function in LNp.75],

Ch 4. #61, #75 (For practice purpose, you must use the law of total expectation and variance decomposition to find the mean and variance, respectively. Other answers are NOT acceptable.), #77, #80, #92 [Hint. Use the law of total expectation (check LNp.67 and LNp.73 for the mgfs of Poisson and gamma, respectively) to find the mgf of α+X, and compare it with the mgf of negative binomial distribution given in LNp.63], #100 [Hint. Apply the δ-method to find the approximate mean and variance.]

Apr 07 sol 周右林, 陳信諺
4

Ch 2. #61,

Ch 4. #20 [Hint: try sum-to-one method], #89 (For practice purpose, you must use moment generating function to show it and to find its mean and variance. Other solutions are not acceptable.)

Ch 5. #1 [Hint: use Chebyshev's inequality and imitate the proof in LNp.96], #5 [Hint: use the theorem: if an→a, then (1+an/n)n→ea. The mgfs of binomial and Poisson distributions are given in LNp.60 and LNp.67, respectively.], #16, #17 [Hint: use CLT], #28

Ch 6. #8 [Hint: You can use mgf to relate the distributions of 2X and 2Y to the chi-square distribution], #11

Apr 14

sol

(corrected)

沈子翔, 曹詠勛
5

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], #19(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 xi ≥ 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.], #50(a)(b) [Hint: Try to use gamma pdf and gamma function given in LN, ch1-6, p.73&74, to find the 1st moment.], #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)θX(m) , then Σ|Xi-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 21 sol 珮雅, 劉必翔
6

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|Xi| or E(Xi2).], #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 answer whether the pdfs form an exponential family, #47(a)(b)(c), also, obtain the Fisher information of the i.i.d. sample and identify the asymptotic sampling distribution of the MLE, #49, #50(c), also, obtain the Fisher information of the i.i.d. sample and identify the asymptotic sampling distribution of the MLE [Hint: Try to use gamma function and gamma pdf given in LN, ch1-6, p.73&74, to find E(Xi2).], #58(a)(b), also, identify the asymptotic sampling distribution of the MLE [Hint: the log-likelihood function is given in LN, CH8, p.24-25 and notice that the marginal distributions are X1~B(n, (1-θ)2)X2~B(n, 2θ(1-θ))X3~B(n, θ2).], #69, #71, also show the pdfs form an exponential family and find a sufficient and complete statistic, #72, also show that the gamma distribution form a 2-parameter exponential family and show that ΠXi and ΣXi are sufficient and complete.

Apr 28 sol 陳信諺, 廖芳翊
7

Homework 7 problem

May 05 sol 沈子翔, 曹詠勛
8

Homework 8 problem

May 14

sol

(corrected)

珮雅, 劉必翔
9

Ch 9. #1, #2, #3, #4(a)(b) [Hint: apply Neyman-Pearson lemma.], #5, also if false, please explain why, #7 [Hint: apply Neyman-Pearson lemma.], #19(b)(c)(d) [Hint: apply Neyman-Pearson lemma.], #29, #30.

May 26

sol

陳信諺, 廖芳翊
10

Homework 10 problem

Jun 02

sol

沈子翔, 曹詠勛
11

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 independent random variables, each distributed as Poisson P(λi), i=1,..., 12, and examine whether λi's are equal], #40 [Hint: p1+p2=1 and X1+X2=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 O1, O2, and O3 be the numbers of AA, Aa, and aa, respectively. Then, (O1, O2, O3)~Multinomial(n,  (1-θ)2, 2θ(1-θ), θ2). The MLE of θ, where 0<θ<1, is (2X3+X2)/(2n) (see LN, CH8, p.25). Use O1, O2, O3 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 H0 and HA (or the differences on their Ω0 and Ω).].

Jun 09

sol

珮雅, 劉必翔

12

textbook (2nd ed.) Ch 15. (Problem statements)

(turn-in questions)

Ch 15. #1, #11, #13, #15, #23, #24, #28, #29.

 

(no-need-to-turn-in question)

Ch 15. #8, #9.

Jun 16

sol

(corrected)

陳信諺, 廖芳翊