ECE 730 Modern Probability Theory and Stochastic Processes, Lec. 1, Fall 2004


Last Modified: Tue 15 Oct 2019, 01:45 PM, CDT

Clock Icon  Instructor Office Hours: MWF 9:00-9:45 and 11:00-11:45 or by appointment.
See what we did in Fall 2003. Fall 2004 will be quite similar.
PC Icon Java Demos for Probability and Statistics (by Prof. Stanton, Cal. State Univ., San Bernardino)
Papers Icon Syllabus
Papers Icon Homework Solutions and Notes from Discussions
Tacked Note Icon Class Schedule for Fall 2004
9/3     Friday.  Started Section 1.2 Review of Set Notation, pp. 9-14.
	Assigned HW #1 Ch 1: 7(d) 8, 9, 27 DUE WED 9/8.

9/6 Labor Day --- NO CLASSES. 9/8 Covered Functions, Countable & Uncountable Sets, pp. 14-18. Axioms & Properties of Probability, pp. 24-26. Assigned HW #2 Ch 1: 12, 14, 29, 30, 35, 43 DUE WED 9/15. 9/10 Covered limit properties (1.13)-(1.17), pp. 26-28. Worked Problems 33 and 34 in class. Covered Note 1 on sigma algebras, pp. 47-48. Defined conditional probability and derived the law of total probability, pp.
9/13 Discussed independence, pp. 32-37. If A & B are indep., so are A & B^c, A^c & B, A^c & B^c. If P(B)=0, it is independent of every event. Similarly if P(B)=1. Independence for many events. Examples 1.24, 1.26, and 1.27. Chapter 2, pp. 70-81. Emphasized Definition of RV. Notation, Example 2.3, Eq. (2.4), pmfs, Poisson RV. Multiple RVs, independent RVs. Example 2.10. 9/15 Example 2.11 max and min problems. Geometric RVs, pp. 82-83. Joint pmfs and independence pp. 86-87. Expectation, Example 2.22. LOTUS eq. (2.14). Moments, variance: (2.16) & (2.17). Example 2.29. Indicator functions, Markov and Chebyshev inequalities, pp. 96-98. Expectation & independence, pp. 99-100. Correlation & Covariance, pp. 100-101. Assigned HW #3 Ch 1: 16, 46; Ch 2: 35, 39, 40, 44 DUE WED 9/22. 9/17 Correlation coefficient; for uncorrelated RVs, variance of a sum is the sum of variances (2.27). Contrast between uncorrelated and independent p. 103. Chapter 3: probability generating function (3.1), factorization property (3.2), obtaining probabilities from G (3.4), moments from G (3.5), Bernoulli and binomial pgfs p. 121, binomial pmf p. 122 - see graphs on p. 123. Substitution Law (3.13), "dropping conditioning" (3.14). Conditional expectation: see (3.21), (3.22), (3.23). You should go over Example 3.21 to see how it's used. Chapter 4: Common densities: uniform, exponential, Laplace, Gaussian. See graphs on pp. 153, 156. Gamma density: See pp. 159-160.
9/20 Derived Law Of The Unconscious Statistician (LOTUS), pp. 167-168. Worked Example 4.14 (Cauchy RV has no mean value). Worked Example 4.11 (Moments of N(0,1) RV). Computed moment generating function of N(0,1) and its characteristic function, Examples 4.15 (first part only) and 4.18. Section 5.5 Properties of CDFs, pp. 217-220. 9/22 Worked/discussed Chapter 5 problems: 10, 11, 16. Assigned HW #4 Ch 4: 13, 14(d), 16(b), 17, 19, 20, 32, 43, 44, 52, 60 DUE WED 9/29. See Student's t densities converge to Gaussian. 9/24 Worked Chapter 5 Problems 18, 19(a)(b)(c), discussed 20 and 21. Chapter 7: Continuous RVs that are not jointly continuous, pp. 309-310.
9/27 For jointly continuous RVs, conditional density (7.12), law of total probability (7.13) and (7.14). Law of substitution (7.15). Worked Example 7.15. Started Chapter 8, Sections 8.1-8.2 pp. 340-346. 9/29 Covered Cross-Covariance, Characteristic Functions, Decorrelation and the Karhunen-Loeve Expansion, pp. 346-350. Started Section 8.3 Linear Estimation of Random Vectors, pp. 350-351. Assigned HW #5 Ch 7: 37, 42; Ch 8: 22, 24, 38, 40 DUE WED 10/6. 10/1 Worked Example 8.9. Covered Section 8.6 Transformations of Random Vectors. Click for a corrected version of Example 8.19.
10/4 Derived the Orthogonality Principle that (8.10) implies (8.9). Then showed that (8.9) holds if and only if A C_Y = C_{X Y} (8.8). In Section 8.5 discussed only MMSE estimation. Derived Orthogonality Principle that (8.14) implies (8.15) and showed that the optimal g(y) = E[X|Y=y]. Started Chapter 9 on Gaussian Random Vectors, pp. 370-374. 10/6 Worked Examples 9.4 and 9.5. Covered Gaussian density in Section 9.4. See level surfaces of a 3-d Gaussian density. Started Section 9.5 Conditional Expectation and Conditional Probability. Announced Exam 1 will be Tuesday evening, Oct. 19 at 7:15 pm. Assigned HW #6 Ch 9: 6, 13, 14, 16, 17 DUE WED 10/13. 10/8 Briefly discussed simulation, p. 376. Covered Section 9.5. Started Chapter 10, pp. 393-399.
10/11 Continued with Chapter 10, pp. 399-409 through eq. (10.18) SY(f) = |H(f)|^2 SX(f). Click here to see examples of correlation functions R(tau) and their processes. Also see corresponding power spectral densites SX(f). Realizations of bandlimited white noise and white noise passed through an RC filter are also shown. 10/13 Discussed Examples 10.15, 10.16, and 10.17. Started Section 10.5 Power Spectral Densities. Worked Example 10.18. Defined White Noise. Recommended that you study Examples 10.20 and 10.21. Covered Section 10.6 Characterization of Correlation Functions. Exam 1 Review Questions: Ch 1: 10, 13, 17-20, 31, 44, 47; Ch 2: 14; Ch 3: 31; Ch 4: 30, 58; Ch 7: 38, 39(b), 56; Ch 8: 20, 21, 27, 28, 41; Ch 9: 18. 10/15 Covered Section 11.1 The Poisson Process; skipped derivation of Poisson probabilities. Briefly mentioned marked point processes and shot noise. Briefly mentioned Section 11.2 Renewal Processes.
10/18 Review for Exam. 10/19 TUESDAY EXAM 1 7:15 pm - ??? in room 3534 EH. The tables of discrete and continuous RVs and Fourier transforms will be included on the exam. You may bring to the exam one 8.5 x 11 in paper with any additional formulas you want. 10/20 Went over exam. Started Section 11.3 on the Wiener Process. Assigned HW #7 Ch 10: 1, 4, 7, 10, 19, 21, 32, 47 DUE WED 10/27. 10/22 Continued Section 11.3: The Wiener Integral and Random Walk Approximation of the Wiener Process. Started Section 11.4.
10/25 Answered questions. Continued Section 11.4. Worked Example 11.9. 10/27 Worked Problems 35-37 in Chapter 11. Started introducing Chapters 13 and 14 on convergence of random variables. Assigned HW #8 Ch 11: 9, 10, 12, 19, 24, 27, 28, 46 DUE WED 11/3. 10/29 Covered Section 13.1, but skipped derivation in Example 13.4. Skipped Example 13.6. Started Section 13.2 through top of p. 532.
11/1 Finished Section 13.2 on Normed Vector Spaces of RVs (skipped subsection on Mean-Square Integrals). Started Section 13.3 on the Karhunen-Loeve Expansion. 11/3 Derived KL expansion, pp. 537-538. Briefly discussed Mercer's Th. on pp. 538-539 and complete orthonormal sets on p. 539. Discussed Examples 13.11, 13.12, and 13.13. Assigned HW #9 Ch 13: 1, 2, 9, 10, 11, 12, 15, 21, 23, 24, 25 DUE WED 11/10. 11/5 Worked Problems 7, 8, 16-19 in Ch. 13. Also showed directly that a convergent sequence is bounded. Covered Section 13.4 The Wiener Integral (Again).
11/8 Covered Section 13.5 Projections, Orthogonality Principle, Projection Theorem. Worked Example 13.14. Skipped Example 13.15. Started Section 13.5 Conditional Expectation and Probability. Briefly discussed Example 13.16. 11/10 Covered the characterization equation of conditional expectation, your choice of (13.23) or (13.26). Showed existence of E[X|Y] when X has finite second moment using the projection theorem. Briefly mentioned how to extend for E[|X|] finite (see p. 549 if you are curious for more details). Skipped Examples 13.18 and 13.19. Worked Examples 13.20 and 13.21. Skipped Example 13.22. Briefly discussed conditional probability onpp. 553-554. Emphasized the SMOOTHING PROPERTY and worked Example 13.23 (The Chapman-Kolmogorov Equation). Assigned HW #10 Ch 13: 37, 41, 45, 46, 52, 53, 56, 57, 60 DUE WED 11/17. 11/12 Worked Problems 58, 59, and 61 in Chapter 13. Covered Section 14.1 Convergence in Probability, but skipped Example 14.2. Started Section 14.3 Convergence in Distribution. Worked Example 14.4.
11/15 Emphasized the 7 things to know about convergence in distribution on p. 601. 1) Showed that convergence in probability implies convergence in distribution. 2) Showed that if X_n converges in distribution to a constant RV, then it coverges in probability. 3) Gave an example of a sequence that converges in distribution but not in probability. 4) Sketched a proof of the fact that if X_n converges in distribution to X, then E[g(X_n)] converges to E[g(X)] for all bounded continuous functions g. (The converse is also true, as shown in the problems.) 11/17 Worked Examples 14.7 and 14.8. Discussed why Wiener integrals are Gaussian RVs and why Y_t = int_0^\infty g(t,\tau) d W_\tau is a Gaussian random process. Discussed the Central Limit Theorem (for more background, see Section 5.6 of Chapter 5). Worked Example 14.10. Motivated Example 14.11 Slutsky's Theorem, but did not work out details. Started Section 14.3 Almost Sure Convergence. Assigned HW #11 Ch 14: 1, 3, 4, 10, 12, 13, 14, 18, 19 DUE WED 11/24. 11/19 Worked Examples 14.12, 14.13, 14.14, and 14.15. Stated SLLN and WLLN. Contrasted WLLN with version in Examples 13.3 and 13.4.
11/22 Worked Example 14.16 and sketched in pictures the idea behind Example 14.17. Started Chapter 12 Markov Chains. Used the smoothing property to solve Examples 12.1 and 12.2. Worked Example 12.3 (Random Walk). Covered derivations on pp. 487-488. 11/24 Worked Problems 6, 32, 34, and 42 in Chapter 14. In Chapter 12, introduced MCs with stationary transition probabilities, transition probability matrices, state transition diagrams. Mentioned examples of random walk, random walk with barrier at the origin. Reflecting and absorbing barriers. Birth-death processes, pure birth processes, pure death processes. Assigned HW #12 Ch 14: 23, 24, 25, 27, 30, 31, 33, 35, 41, 43, 46 DUE WED 12/1. 11/26 No class -- Thanksgiving Recess.
11/29 Consequences of time homogeneity, m-step transition probabilities, Chapman-Kolmogorov equation, stationary/equillibrium distributions. Worked Examples 12.4 and 12.5. 12/1 Discussed Example 12.6. Started Section 12.3 Limit Distributions, covering pp. 497-501. Assigned HW #13 Ch 12: 5, 9, 10 DUE WED 12/8. Strongly Recommended Problems: Do BY HAND 6, 7, 8. Also solve numerically --- see 11 and 12. Click on problem numbers to see code, which you can then "Save As". 12/3 Covered First Entrance Times, Example 12.10, Number of Visits to a State pp. 501-506.
12/6 Reviewed pp. 504-505. Covered pp. 506-508: Classes of States, communicating states, class properties, period of a state. 12/8 Summarized relationships between limiting n-step transition probabilities and stationary distributions. Showed that if two states communicate, then they have the same period. REVIEW QUESTIONS: Ch 10: 11, 14, 31; Ch 11: 11, 25, 30, 44; Ch 12: 14; Ch 13: 13, 14, 26, 29, 38, 54, 62, 63; Ch 14: 8, 15, 16, 44, 45, 48. 12/10 Covered Section 12.4 Continuous-Time Markov Chains, pp. 510-513. Only quickly stated Kolmogorov's forward and backward equations and stationary distribution formula on p. 515.
12/13 Worked review problems. Distributed teaching evaluations. 12/15 Last Class Day. Worked review problems. 12/18 SATURDAY: FINAL EXAM 12:25-2:25 in 3534 EH. You may bring to the exam two 8.5 in. x 11 in. sheets of paper with any formulas you want. Since the following tables of pmfs, pdfs, transforms, and series will be included on the exam, you don't need to write these down or memorize them.

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