ECE 331 Introduction to Random Signal Analysis and Statistics, Lec. 1, Spring 2007

1/22	Monday.  Course overview.  Covered Section 1.1 Sample Spaces,
	Outcomes, and Events.  Covered Section 1.2 Review of Set Notation
	through the beginning of Example 1.4.

1/24	Finished Example 1.4.  Discussed Partitions.  Skip subsections on
	Functions and Countable and Uncountable Sets.  Covered Section 1.3
	Probability Models: Examples 1.10, 1.12, 1.15-1.16.  Skip Examples
	1.11, 1.13, and 1.14.
	Assigned HW#1 Ch 1: 2, 5, 7, 23, 26, 27 DUE WED 1/31.

1/26	You should look over Example 1.17.  Covered Section 1.4 Axioms and
	Properties of Probability.  Started Section 1.5 Conditional Probability.
	Worked Example 1.20.

1/29	Worked Example 1.21.  Covered Section 1.6 Independence through
	Example 1.24.

1/31	Worked Examples 1.25-1.27.  Covered Chapter 2 through Example 2.2.
	Please read the rest of p. 65 on notation.
	Assigned HW#2 Ch 1: 54, 55, 58, 62, 64, 65 DUE WED 2/7.

2/2	Reviewed RV notation on p. 65.  Worked Examples 2.3 and 2.4.  Started
	Section 2.2 Discrete RVs, Integer-Valued RVs, Probability Mass
	Functions.  The Uniform RV, Example 2.6, The Poisson RV.

2/5	Worked Example 2.7  Started Section 2.3 Multiple RVs, Independent RVs,
	worked Examples 2.8-2.11.

2/7	The Geometric Random Variables, Examples 2.12, 2.13.  Joint pmfs,
	marginal pmfs.  Skip Computing Probabilities with Matlab.  Start
	Section 2.4 Expectation.
	Assigned HW#3 Ch 2: 11, 14, 17, 18, 26(a)(b), 32, 35 DUE WED 2/14.

2/9	Worked Examples 2.21, 2.22.  Skip Examples 2.23-2.25.  LOTUS-skip
	derivation.  Linearity of expectation.  Moments.  Worked Example 2.27.
	Variance formula (2.17).  Worked Example 2.28.

2/12	Worked Example 2.29.  Introduced indicator functions.  Worked
	Example 2.31.  Introduced the Markov inequality (2.18) and its
	generalization (2.19).  You may wish to read Examples 2.32 and 2.33
	on your own to see how these bounds can be used.  Derived (2.19),
	the Chebyshev inequality (2.21) and (2.22).  You may wish to read
	Example 2.34 on your own to see an application of the Chebyshev
	inequality.  Showed that if X and Y are independent discrete RVs,
	then for any functions h(x) and k(y), E[h(X)k(Y)]=E[h(X)]E[k(Y)].
	Defined correlation E[XY] and correlation coefficient ρXY.

2/14	Worked Example 2.35.  For uncorrelated RVs, showed that the variance
	of the sum is the sum of the variances (2.28).  Started Chapter 3,
	Section 3.1 Probability Generating Functions.  Worked Example 3.2.
	Derived (3.4) for obtaining probabilities from the pgf and (3.5)
	for obtaining moments from the pgf.  Worked Example 3.4 Started
	Section 3.2 The Binomial RV.  Discussed Example 3.5.
	Assigned HW#4 Ch 2: 37, 45,
	Ch 3: 2, 5, 8, 11, 14  DUE WED 2/21.

2/16	Used pgfs to find the pmf of the sum of i.i.d. Bernoulli(p) RVs.
	This sum is binomial(n,p).  Covered Section 3.3 The Weak Law of
	Large Numbers.


2/19	Started Section 3.4 Conditional Probability.  Introduced conditional
	pmfs.  Worked Examples 3.10-3.13 and first part of 3.15.

2/21	Finished Example 3.15.  Worked Example 3.16.  Introduced the
	Substitution Law.  Worked Example 3.17.
	Suggested Exam 1 Review Problems:
	Ch 1: 53, 56, 57, 59, 63, 66, 67.
	Ch 2: 15, 19, 20, 25, 34, 36.
	Ch 3: 12, 13.

2/23	Worked Example 3.18.  Started Section 3.5 Conditional Expectation.
	Worked Examples 3.19 and 3.20.  Introduced Laws of Substitution and
	Total Probability for conditional expectation.

2/26	Worked Example 3.21.  Started Ch 4.  Discussed uniform pdf.  Skipped
	Examples 4.1 and 4.2, but you should work them on your own.
	Discussed exponential pdf and worked Example 4.3.

2/28	Exam 1 --- In class.
	Assigned HW#5 Ch 3: 23, 27, 30, 31, 41, 42 DUE WED 3/7.

3/2	Solved Exam 1.  Worked Examples 4.1, 4.2, and 4.4.  Discuessed the
	"Paradox of Continuous RVs" on p. 149; i.e., for a continuous RV,
	P(X=b)=0 for any value b.

3/5	Introduced the Cauchy and Gaussian densities.  Skipped example 4.5,
	but you may wish to do it yourself.  Introduced location and scale
	parameters and the family of gamma densities.  Started Section 4.2
	Expectation of a Single RV.  Used integration by parts to compute
	the mean of an exp(lambda) RV.

3/7	Worked Examples 4.7, 4.9-4.12.  Started Section 4.3 Transform Methods.
	Worked first part of Example 4.15.
	Assigned HW#6 Ch 4: 4(a), 7, 8 (y=λxp), 12, 13, 14(a)(b), 23, 29 DUE WED 3/14.

3/9	Worked Examles 4.16-4.18.  Started Characteristic Functions.  Worked
	Examples 4.19-4.21.  Started Section 4.4 Expectation of Multiple RVs.
	Worked Example 4.22.

3/12	Worked Examples 4.23 and 4.24.  Worked Problems 44(a), 45, and 31
	in Chapter 4.

3/14	Started Chapter 5, Section 5.1.  Worked Examples 5.1-5.3.  Discussed
	bounding the Gaussian complementary cdf, Q(x) as in Problem 22 of Ch. 5.
	Assigned HW#7 Ch 4: 32, 40, 46, 54, 55
	              Ch 5: 7, 8 DUE WED 3/21.

3/16	Worked Examples 5.5-5.9.  Section 5.3 through the top of p. 199.

3/19	Briefly discussed cdfs of integer-valued, and then discrete RVs.
	Such cdfs are piecewise constant, and the jump in the cdf at x_j
	is P(X=x_j).  Discussed the 8 properties of cdfs in Section 5.3.
	Started Chapter 7.  Focused on the definition of Cartesian product
	sets and how they can be used to describe probabilites involving
	only a single RV --- see pp. 289-290.  Introduced the joint cdf
	and the rectangle formula on p. 291.  See notes on computing
	probabilities for pairs of RVs.

3/21	Introduced marginal cdfs eqs. (7.3) and (7.4) on p. 292.  Worked
	Example 7.7.  Introduced joint cdfs of independent RVs.  Started
	Section 7.2 Jointly Continuous RVs.  Worked Example 7.12.
	Assigned HW#8 Ch 5: 11, 16, 19
	              Ch 7: 30, 32(a), 33(a)(b), 34 DUE WED 3/28.

3/23	Worked Example 7.13.  Expectation for pairs of jointly continuous RVs.
	Started Section 7.3 Conditional Probability and Conditional Expecation.
	Worked Example 7.14.

3/26	Worked Example 7.15.  Worked Problems 31 and 32(b) in Ch 7.  Discussed
	Conditional Expectation.  Worked first part of Example 7.18.

3/28	Finished Example 7.18.  Worked another example.  Started Chapter 10.
	Discussed Figs. 10.1-10.7.
	Suggested Exam 2 Review Problems:
	Ch 4: 30(a)(b), 31, 34, 35, 56, 57, 59-64.
	Ch 7: 33(c)(d)(e), 36, 37, 39(b)(d), 40, 41, 59.

3/30	Started Section 10.2.  Introduced mean function, (auto)-correlation
	function, covariance function.  Worked Example 10.8, 10.11.
	Introduced cross-correlation function.  Worked Example 10.12, from
	which (10.10) and (10.11) are immediate.

4/2	No class --- Spring recess.
4/4	No class --- Spring recess.
4/6	No class --- Spring recess.

4/9	Worked review problems.

4/11	Exam 2 --- In class.
	Assigned HW#9 Ch 10: 8, 17(use table), 18(use table), 24,
	28, 29 DUE WED 4/18.

4/13	Section 10.3 and part of Section 10.4.

4/16	Finished Section 10.4, started Section 10.5.

4/18	Finished Section 10.5.
	Assigned HW#10 Ch 10: 22, 36, 37, 39 WED 4/25.

4/20	No class ---  Engineering EXPO http://engineeringexpo.wisc.edu/

4/23	Went over Exam 2.

4/25	Section 10.6 - first part only, NOT the subsection on correlation
	functions of deterministic waveforms.  Covered Section 10.7 The
	Matched Filter.
	Assigned HW#11 Ch 10: 38, 40, 41, 47, 49, 50 WED 5/2.

4/27	Derived the Cauchy-Schwarz inequality for random variables (2.24).
	Compared it with the waveform version used to derive the Matched
	Filter (see Problem 2 in Chapter 10).  Covered Section 10.8 The Wiener
	Filter.

4/30	Discussed interpretation of the Wiener filter when U_t = V_t + X_t,
	where V_t and X_t are J-WSS with V_t and X_t uncorrelated.  Started
	Section 11.1 The Poisson Process.  Worked Example 11.1.

5/2	Finished Section 11.1 through Example 11.2.  Covered Section 5.6 The
	Central Limit Theorem up to, but not including Example 5.17.  Also
	covered the dervation of the CLT on pp. 214-215. 
	Assigned HW#12 Ch 10: 55; Ch 11: 1, 3, 4, 8, 9 WED 5/9.

5/4	Covered Section 6.1 through Example 6.2.  Distributed teaching
	evaluations.

5/7	Covered Section 6.3 Confidence Intervals -- Known Mean.  Covered
	Section 6.4 Confidence Intervals -- Unknown Mean.

5/9	Reviewed confidence intervals, Sections 6.3 and 6.4.
	Finished Section 6.1.  Discussed histograms, Section 6.2.
	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 10: 42
	Ch 11:  5,  6, 10, 11.
	Work review problems for previous midterms, work old HWs.
5/11

5/16	Final Exam --- 7:25 pm Wed. MAY 16 in 2317 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.