| Homework | Question | Due Day | Solution | Grader |
| 1 |
Ch 2. #40, #70. Ch 3. #10, #21, #25 [Hint: Let S=±1 with probability 1/2 each and S independent of X. Then, Y and WX have same distribution], #53 [Hint: Here is one way to get the answer. Let Y1=U1U2/(U32), Y2=U1, Y3=U2. Then, you can use Theorem2.7 in LNp.2-32 to get the joint pdf of Y1, Y2, and Y3. Note that the joint pdf is zero outside the region 0<Y2<1, 0<Y3<1, Y2Y3<Y1<∞. Use the joint pdf to obtain P(Y1<1).], #77. Ch 4. #26, #49, #59 [Hint: the joint pdf of (X, Y) is given in textbook p.80, Example E], #61, #72. |
Mar 10 | Sol | 呂心樂、徐瑋崙 |
| 2 |
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 k≥k0 and p(k)/p(k+1)<1 for any k<k0, where p(·) is the pmf of Poisson.], #44, #55 [Hint: You can use the Table 2 in textbook Appendix B, p.A7], #61 [Hint: You can show it by using mgf]. Ch 3. #22 [Hint: You can first show that: If X1~Poisson(λ1), 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)).], #47. Ch 4. #20 [Hint: try sum-to-one method], #42 [Hint: Use the cdf, mean, variance equations given in LNp.2-71 to find the probability], #79, #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.), #99 [Hint: use δ method]. |
Mar 17 | Sol | 侯秉逸、李友棣 |
| 3 |
(turn-in questions) Ch 5. #1 [Hint: use Chebyshev's inequality and imitate the proof in LNp.2-93], #3, #5 [Hint: use the theorem: if an→a, then (1+an/n)n→ea. The mgfs of binomial and Poisson are given in LNp.2-60 and LNp.2-67, respectively.], #10, #11, #17 [Hint: use CLT], #18 [Hint: Let Xi be the weight of the ith package, i=1,...,100. Use CLT to find P(ΣXi > 1700).], #21 [Hint: (i) This is a generalization of the Monte Carlo integration given in Ex5.3, LNp.2-93; (ii) For (c), by Cauchy-Schwarz inequality, we have
], #23 [Hint: This is an application of LLN and the Monte Carlo integration given in Ex5.3, LNp.2-93], #28. Ch 6. #8 [Hint: You can use mgf to relate the distributions of 2X and 2Y to the chi-square distribution], #11.
(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 24 | Sol |
蘇羿豪、蔡采陵 |
| 4 |
Ch 8. #7(a)(b) [Hint: the 1st moment of geometric distribution is given in LN, ch1-6, p.2-62], #9 [Hint: check the diagram in LNp.3-12], #16(a)(b) [Hint: The 1st moment is zero, and you can use gamma pdf and gamma function given in LN, ch1-6, p.2-73&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.2-68, to model the data.], #50(a)(b) [Hint: Try to use gamma pdf and gamma function given in LN, ch1-6, p.2-73&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)≥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.2-73&74] |
Apr 02 | Sol | 呂心樂、徐瑋崙 |
| 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.2-62], #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.2-73&74, to find E|Xi| or E(Xi2).], #47(a)(b)(c), also, obtain the Fisher information of the i.i.d. sample and identify the asymptotic sampling distribution of the MLE, #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.2-73&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 LNp.3-24 and notice that the marginal distributions are X1~B(n, (1-θ)2), X2~B(n, 2θ(1-θ)), X3~B(n, θ2).]. |
Apr 14 | Sol | 侯秉逸、李友棣 |
| 6 |
Ch 8. #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; #49; #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 21 | Sol | 蘇羿豪、蔡采陵 |
| 7 | Apr 28 |
