ECE 730 Modern Probability Theory and Stochastic Processes, Lec. 1, Fall 2004
Last Modified: Tue 15 Oct 2019, 01:45 PM, CDT
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.
Java
Demos for Probability and Statistics
(by
Prof.
Stanton,
Cal. State Univ., San Bernardino)
Syllabus
Homework Solutions and Notes from Discussions
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.
Web Page Contact: John (dot) Gubner (at) wisc (dot) edu