| 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 | ||
