ECE 331 Introduction to Random Signal Analysis and Statistics, Lec. 1, Fall 2006

9/5	Tuesday.  Course overview.  Covered Section 1.1 Sample Spaces,
	Outcomes, and Events.  Covered Section 1.2 Review of Set Notation
	through Partitions.  Skip subsections on Functions and Countable and
	Uncountable Sets.  Started Section 1.3 Probability Models.

9/7	Worked Examples 1.10, 1.12-1.17.  Started Section 1.4 Axioms and
	Properties of Probability through through p. 24.
	Assigned HW#1 Ch 1: 2, 4, 5, 6, 7 DUE THU 9/14.

9/12	Derived limit properties of probability.  Stated the union bound.
	Covered Section 1.5 Conditional Probability: law of total probability,
	Bayes' rule, Examples 1.20, 1.21.  Generalize Bayes' rule.  Started
	Section 1.6 Independence: defined independent events.

9/14	Finished Section 1.6 Independence.  Worked Problems 56, 58, and 67.
	Assigned HW#2 Ch 1: 23, 24, 26, 53, 54, 55, 62, 63, 65 DUE THU 9/21.

9/19	Covered Ch. 2 Introduction to Discrete RVs up to Example 2.7.

9/21	Begin with Example 2.7.  Covered Section 2.3 Multiple Random Variables
	up to Example 2.14.  Skip Derivations subsections.  Skip Computing
	Probabilities with Matlab.
	Assigned HW#3 Ch 2: 1, 2, 7, 8, 10, 11, 14, 17, 18 DUE THU 9/28.

9/26	Worked Example 2.14.  Started Section 2.4 Expectation.  Worked Examples
	2.20, 2.21, 2.24.  Discussed conventions regarding infinity.  Stated
	Law of the Unconscious Statistician (LOTUS) for one RV  and for
	pairs of RVs.  Showed that expectation is LINEAR.  Defined moments,
	variance, standard deviation.  Worked Examples 2.26 and 2.27.
	Derived the VARIANCE FORMULA: var(X) = E[X2] - (E[X])2.

9/28	Worked Examples 2.28-2.30.  Introduce INDICATOR FUNCTIONS.  Derived
	Markov and Chebyshev inequalities.  Worked Examples 2.31-2.33.
	You might want to try to work Example 2.34 yourself.
	Expectation & Independence.  Correlation and covariance.  The
	Cauchy-Schwarz inequality, correlation coefficient, uncorrelated,
	covariance.
	Assigned HW#4 Ch 2: 23, 32, 33, 34, 35, 37 DUE THU 10/5.

10/3	Worked Example 2.39.  Showed that for uncorrelated RVs, the variance
	of the sum is the sum of the variances (p. 95.).  Started Chapter 3.
	Covered Section 3.1 except for the examples.

10/5	Worked Examples 3.3 and 3.4.  Used pgfs to show that the sum of
	n i.i.d. Bernoulli(p) RVs is binomial(n,p).  Stated the binomial
	theorem, Pascal's triangle.  Discussed "n choose k" terminology.
	Derived the Poisson approximation of binomial probabilities.
	Started Section 3.4 Conditional Probability.  Worked Examples
	3.11 and 3.12.
	Exam 1 Review Problems:
	Ch 1: 27, 57, 59, 64, 66, 67
	Ch 2: 9, 15, 19, 20, 25, 36, 45

10/10	Worked review problems.  Discussed linear estimation as in Problems
	38 and 43 in Chapter 2; connected these problems with the orthogonality
	principle.

10/12	EXAM 1 IN CLASS. Chapters 1 & 2 ONLY.
	Assigned HW#5 Ch 3: 1, 2, 5, 8, 11, 14 DUE THU 10/19.

10/17	Introduced Law of Total Probability for discrete RVs, pp. 120-121.
	Worked Examples 3.13, 3.15, and 3.16.  Introduced the Substitution
	Law pp. 122-123.  Essentially worked Example 3.17.

10/19	Worked Example 3.18.  Covered Section 3.5 Conditional Expectation.
	Started Chapter 4, discussed uniform, exponential, Laplace, Cauchy,
	and Gaussian densities.
	Assigned HW#6 Ch 3: 23, 27, 30, 31, 41, 42  DUE THU 10/26.

10/24	Worked Examples 4.1-4.5.  Showed that Gaussian density integrates
	to one.  Discussed location and scale parameters and gamma densities.
	Worked Examples 4.7 and 4.9.

10/26	Worked Examples 4.10-4.14.  Section 4.3 Transform Methods: Moment
	generating functions and characteristic functions.  Worked
	Examples 4.15-4.18, 4.22-4.23.
	Assigned HW#7 Ch 4: 1, 4(a), 6, 7, 8 (y=λxp), 12, 13, 14(a)(b) DUE THU 11/2.

10/31	Worked Examples 4.20-4.21, 4.24-4.25, 4.28.
	Started Chapter 5, Section 5.1, worked Examples 5.1, 5.2, 5.5.

11/2	Worked Problems 44(a) and 45(a) in Ch 4.  Worked Example 5.6-5.8. 
	Derived Leibniz' rule for differentiating integrals.  Worked
	Example 5.9.  Showed how to use a uniform RV X to produce a RV Y
	with desired cdf F(y).  Section 5.2: Briefly discussed cdfs of
	discrete RVs.
	Assigned HW#8 Ch 4: 23, 29, 32, 40, 46, 54, 55 DUE THU 11/9.
	

11/7	Section 7.1: Skipped all examples.  Key points (pp. 289-291): Product
	Sets and Marginal Probabilities, Joint CDFs, the Rectangle Formula.
	p. 292: How to obtain marginal cdfs from the joint cdf: Formulas (7.3)
	& (7.4).  pp. 294-295: RVs X and Y are independent if and only if
	FXY(x,y)=FX(x)FY(y).
	Section 7.2: Jointly Continuous RVs, Skipped all examples.  Key formulas:
	(7.9) joint pdf = mixed partial of joint CDF.  (7.10) (and similar
	boxed formula below it): marginal density is obtained by integrating
	out the variable you don't want.  Jointly continuous RVs X and Y
	are independent if and only if fXY(x,y)=fX(x)fY(y).
	Review Problems
	Ch 3: 12, 13, [28, 29], 39
	Ch 4: 5, 30(a)(b), 33, [34, 35], [56, 59], 57, [60-64]
	Ch 5: 7, 8, 11, 16, 19

11/9	In class worked Ch 3 Problem 45, Ch 4 Problem 31, Ch 5 Problems 13,
	15, and 21(a).  Section 7.3: Introduced a limit definition of
	conditional probability for jointly continuous random variables
	and showed that it satisfies the Law of Total Probability.  Worked
	Example 7.14.


11/14	Exam 2 -- in class.

11/16	Went over Exam 2.  Worked Example 7.15 and Ch 7 Problems 30,
	32(b), and 39(c).
	Assigned HW#9 Ch 7: 26, 31, 32(a), 33(a)(b), 34, 36, 40 DUE THU 11/30.

11/21	Started Chapter 10.  Covered Sections 3.1, 3.2, and started Section 3.3
	but skipped Examples 10.9, 10.10, 10.13, 10.14.

11/23	THANKSGIVING -- NO CLASS

11/28	Discussed Examples 10.15-10.18.  Covered Transforms of Correlation
	Functions, Examples 10.20-10.22.  Covered first remark on p. 400.
	Section 10.4 WSS Processes Through LTI Systems: Key formulas are
	(10.22) and (10.23).  Note also the definition of Joint WSS in
	the 2nd paragraph on p. 402.  Worked Example 10.23.  Covered first
	paragraph of Section 10.5 Power Spectral Densities for WSS Processes.

11/30	Continued discussion of power in process.  For WSS processes, there
	are 3 formulas for the expected time-average power (10.24).  Showed
	that the integral of SX(f) over a frequency band gives the power
	in the band.  That's why it's called the power spectral density.
	Defined white noise, discussed bandlimited white noise as shown
	in Fig. 10.14.
	Assigned HW#10 Ch 10: 8, 17(use table), 18(use table), 24, 28,
	29, 33, 36 DUE THU 12/7.

12/5	White noise through an RC filter.  Worked Examples 10.26-10.28.
	Section 10.6 Characterization of Correlation Functions; we are
	NOT covering the subsection Correlation Functions for Deterministic
	Signals.  Section 10.7 The Matched Filter ONLY through Example 10.31.

12/7	Section 10.8 The Wiener Filter - ONLY through (10.37).
	Distributed teaching evaluations.
	Assigned HW#11 Ch 10: 37, 39, 40  DUE THU 12/14.

12/12	Section 11.1 The Poisson Process - only through Example 11.2.
	Section 5.6 The Central Limit Theorem - up to but not including
	Example 5.17.  Also skip Example 5.16.  Focused statement of the
	theorem (gray box on p. 208) and on derivation of (5.8).

12/14	Worked problems from Chapter 10.


12/22	FRIDAY, Final Exam, 12:25 pm in 3444 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.

	Review Problems for Final Exam:
	Ch 2: 12, 13
	Ch 3:  3,  7, 18, 32, 43, 46
	Ch 4: 48, 58
	Ch 5:  6,  9, 14
	Ch 7: 33(c)(d)(e), 37, 39(b)(d), 41, 59
	Ch 10: 38, 41, 42