| Homework | Question | Due Day | Solution | Grader |
| 1 |
Ch 2. #67 Ch 3. #7, #10, #21, #64, #66 [Hint: Let {T1, T2}, {T3, T4}, and {T5, T6} be the lifetimes of the top two, middle two, and bottom two components, respectively. Then, the system's lifetime is given by max{min(T1, T2), min(T3, T4), min(T5, T6)}.], #70 Ch 4. #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.] |
Mar 04 | Sol | 林宸緯、馬翌翔 |
| 2 |
(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 X1~P(λ1), X2~P(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. #20 [Hint: try sum-to-one method], #42 [Hint: Use the cdf, mean, variance equations given in LNp.71 to find the probability], #76, #80, #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], #100 [Hint. Apply δ-method to find the approximate mean and variance.].
(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 11 | Sol (corrected) | 孫利東、高童玄 |
| 3 |
Ch 5. #1 [Hint: use Chebyshev's inequality and imitate the proof in LNp.93], #2 [Hint:
], #4, #5 [Hint: use the theorem: if an→a, then (1+an/n)n→ea. 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 18 | Sol | 劉馨隃、陳家桓 |
| 4 |
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 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.], #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|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]. |
Mar 27 | Sol | 林宸緯、馬翌翔 |
| 5 |
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(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 X1~B(n, (1-θ)2), X2~B(n, 2θ(1-θ)), X3~B(n, θ2). |
Apr 08 | Sol | 孫利東、高童玄 |
| 6 |
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 ΠXi and ΣXi are sufficient and complete. |
Apr 15 | Sol | 劉馨隃、陳家桓 |
| 7 | Apr 22 | Sol | 林宸緯、馬翌翔 | |
| 8 | May 01 | Sol | 孫利東、高童玄 | |
| 9 |
Ch 9. #1, #2, #3, #4(a)(b) [Hint: apply Neyman-Pearson lemma.], #5, also if false, please explain why, #21, #29. |
May 13 | Sol | 劉馨隃、陳家桓 |
| 10 | May 20 | Sol | 林宸緯、馬翌翔 | |
| 11 |
Ch 9. #12, #22, #24
#26, also if false, please explain why. #40 [Hint: p1+p2=1 and X1+X2=n], #41 [Hint: Under Ω0, the MLE of pi is Σi Xi/N, where N=Σi ni, for i=1, 2, ..., m. Under Ω, the MLE of pi is Xi/ni, i=1, 2, ..., m.], #43 [Hint: you can use GLR test statistic or Pearson's Chi-square for goodness-of-fit test.], #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 with 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).]. |
May 27 | Sol | 孫利東、高童玄 |
| 12 |
textbook (2nd ed.) Ch 15. (Problem statements) #1, #6, #9, #11, #13, #15, #23, #24, #29. (blue-colored problems are not included in final exam). |
此作業不計分 無需繳交 (no need to turn in this HW) |
Sol | 劉馨隃、陳家桓 |
,