A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z | AA | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | |||||||||||||||||||||||||||

2 | Links: | Slides | https://robertmarks.org/Classes/EE5345-Slides/Slides.html | ||||||||||||||||||||||||

3 | Sylabus | https://robertmarks.org/Classes/ENGR%205345/5345-Syllabus.pdf | |||||||||||||||||||||||||

4 | Text | https://www.amazon.com/gp/product/B00ALTS9TY/ | |||||||||||||||||||||||||

5 | This Link | https://docs.google.com/spreadsheets/d/1Zueo5Zn2TJcLdTBLd9H1U8eWDX8wO-FT_OxTsejiRoY/edit?usp=sharing | |||||||||||||||||||||||||

6 | |||||||||||||||||||||||||||

7 | ECS 5354 | ||||||||||||||||||||||||||

8 | 1 | January 19 | Tues | History | Solve the unfinished game problem | https://youtu.be/FMmsinC9q6A | |||||||||||||||||||||

9 | Due Jan 21 | ||||||||||||||||||||||||||

10 | 2 | January 21 | Thurs | Review | Solve the Deal or No Deal Problem | https://youtu.be/-OqhMVIrJOI | |||||||||||||||||||||

11 | Solve the 5 presidents problem | ||||||||||||||||||||||||||

12 | Due Jan 28 | ||||||||||||||||||||||||||

14 | 3 | January 28 | Thurs | Review | Chapt 2- p89: 77,80 ;p92:104,105 | https://youtu.be/m0q7QDDz5-0 | |||||||||||||||||||||

15 | Chapt 4 - p218: 26 | ||||||||||||||||||||||||||

16 | Convert the MatLab erf to my erf. SOLUTION: | https://docs.google.com/document/d/1zPn3xWldlzPtqKMds98j02e01jvNvkDDp8vqK1OV0D4/edit?usp=sharing | |||||||||||||||||||||||||

17 | Due Feb 4 | ||||||||||||||||||||||||||

18 | 4 | February 2 | Tues | Distributions | https://youtu.be/9RezsChrzaU | ||||||||||||||||||||||

19 | Info Theory | ||||||||||||||||||||||||||

20 | 5 | February 4 | Thurs | Info Theory | Evaluate optimal "20 questions" queries for: | https://youtu.be/c7EmERRdq1I | |||||||||||||||||||||

21 | p = 1/3, 1/3, 1/3 and p = 1/8, 3/8, 1/10, 2/10, 2/10 (hard) | ||||||||||||||||||||||||||

22 | Evaluate E(Questions) for each and 2^(-length_n) for each | ||||||||||||||||||||||||||

23 | Compare to the entropy in each case and comment. | ||||||||||||||||||||||||||

24 | Chapt 3 - p132: 13 | ||||||||||||||||||||||||||

25 | Due Feb 11 | ||||||||||||||||||||||||||

26 | 6 | February 9 | Tues | RV Transformation | https://youtu.be/hPVmeQri_rc | ||||||||||||||||||||||

27 | 7 | February 11 | Thurs | Functions of a RV | https://youtu.be/AAG-h8rh5BA | ||||||||||||||||||||||

28 | HOMEWORK: | Show g(x) = F_X transforms a continuous f_X RV to uniform RV | |||||||||||||||||||||||||

29 | (a) What if X is a discrete RV? (b) a Posson RV? | ||||||||||||||||||||||||||

30 | Show inverse of F_X transforms a continuous uniform RV to an f_X RV | ||||||||||||||||||||||||||

31 | What if X is a discrete RV? | ||||||||||||||||||||||||||

32 | Find f_Y when X is uniform on (-pi/2,pi/2) and g(x)=tan(x) | ||||||||||||||||||||||||||

33 | Derive f_Y if X is zero mean Gaussian and (a) Y=|X|, (b) Y=X^2 | ||||||||||||||||||||||||||

34 | Find g(x) to transform a centered Cauchy to a uniform RV | ||||||||||||||||||||||||||

35 | Due Feb 25 | ||||||||||||||||||||||||||

38 | 8 | February 23 | Tues | Characteristic Functions | https://youtu.be/skA9oyKv9fg | ||||||||||||||||||||||

39 | HOMEWORK: | ||||||||||||||||||||||||||

40 | Generate a bunch of Y from a uniform distribution X using a transformation Y=g(X) | ||||||||||||||||||||||||||

41 | Derive f_Y (y) | ||||||||||||||||||||||||||

42 | Generate a histogram of Y's, nomalize to unit area and compare to f_Y (y) | ||||||||||||||||||||||||||

43 | Due March 4 | ||||||||||||||||||||||||||

44 | 9 | February 25 | Thurs | Characteristic Functions | https://youtu.be/bC9Z9Ls5-bk | ||||||||||||||||||||||

45 | HOMEWORK: | ||||||||||||||||||||||||||

46 | What is the distribution when the RV's from the following RV's are added: | ||||||||||||||||||||||||||

47 | SIx geometric RV's all with given parameter p. | ||||||||||||||||||||||||||

48 | Six Bernoulli RV's all with paramenter p | ||||||||||||||||||||||||||

49 | Six Pascal Negative Binomial RV's all with parameter p but different r's | ||||||||||||||||||||||||||

50 | Three gaussians all with different means and variances | ||||||||||||||||||||||||||

51 | What is the mean and variance when the characteristic function is exp(- |omega| ^N) * exp(-j omega). N is an integer. | ||||||||||||||||||||||||||

52 | Due March 4 | ||||||||||||||||||||||||||

53 | 10 | March 2 | Tues | 2D RV's | https://youtu.be/hWINB89ezfQ | ||||||||||||||||||||||

54 | 11 | March 4 | Thurs | 2D RV's | https://youtu.be/UZ68vjyesSc | ||||||||||||||||||||||

55 | HOMEWORK: | ||||||||||||||||||||||||||

56 | 1. Chose a causal pdf (= 0 for negative x). Plot Pr[X>=a] versus E[X]/a as a function of a and verify Markov's inequality. | ||||||||||||||||||||||||||

57 | 2. Chose a pdf. Plot Pr[ |X-E(X)|>a] versus var(X)/a^2 as a function of a and verify Chebyshev's inequality | ||||||||||||||||||||||||||

58 | 3. Copy the 10x10 histogram on Sheet 2 of this spreadsheet. | ||||||||||||||||||||||||||

59 | (a) Calculate and generate a 2D plot of the corresponding empirical pdf. Make sure to have a number in each location of the 10x10 grid. | ||||||||||||||||||||||||||

60 | (b) Calculate and generate a 2D plot of the corresponding empirical cumulative distribution function | ||||||||||||||||||||||||||

61 | (c) Repeat (a) and (b) when we are given that | ||||||||||||||||||||||||||

62 | (i) 3<=X<=6 and 4<=Y<=6 | ||||||||||||||||||||||||||

63 | (ii) X<Y | ||||||||||||||||||||||||||

64 | Due March 11 | ||||||||||||||||||||||||||

65 | 12 | March 9 | Tues | 2D RV's | https://youtu.be/V4PJUB5oKgc | ||||||||||||||||||||||

66 | 13 | March 11 | Thurs | MD RV's | https://youtu.be/KvhONWbthtk | ||||||||||||||||||||||

67 | MIDTERM: | ||||||||||||||||||||||||||

90 | 14 | March 16 | Tues | MD RV's | https://youtu.be/XyDJnegwzR8 | ||||||||||||||||||||||

91 | 15 | March 18 | Thurs | Law of Large Numbers | https://youtu.be/pj8xdnSBcVw | ||||||||||||||||||||||

92 | Homework | ||||||||||||||||||||||||||

93 | 1. Computer work: Randomly generate points in square. Inside the square is an inscribed circle. | ||||||||||||||||||||||||||

94 | (a) What is the probability a point will be in the circle? | ||||||||||||||||||||||||||

95 | (b) As each new point is added, keep track of and plot the percent of points in the circle divided by the thal number of points chosen. | ||||||||||||||||||||||||||

96 | (c) Does the limit of your plot approach your answer in (a)? | ||||||||||||||||||||||||||

97 | 16 | March 23 | Tues | Central Limit Theorem | https://youtu.be/4AvCvR6SJyg | ||||||||||||||||||||||

98 | Homework | ||||||||||||||||||||||||||

99 | 1. Computer work: Generate a bunch of empirical pdf's. For at least one, use a random number generator. Convolve these pdf's. | ||||||||||||||||||||||||||

100 | (a) Does the convolution have an area of one? | ||||||||||||||||||||||||||

101 | (b) Plot the convolution and the approximated gaussian in the same figure. | ||||||||||||||||||||||||||

102 | 2. 1000 dollar amounts are rounded UP to the nearest dollar. | ||||||||||||||||||||||||||

103 | (a) Estimate the total probability the error is greater that $495. Give a number. | ||||||||||||||||||||||||||

104 | (b) Greater that $1000 dollars? Give a number. | ||||||||||||||||||||||||||

105 | 17 | March 25 | Thurs | Confidence Intervals | https://youtu.be/oVTWKN5EVII | ||||||||||||||||||||||

106 | 1. A roulette wheel has 37 slots. Two of them are green. | ||||||||||||||||||||||||||

107 | (a) Out of 10,000 tries, a ball rolls into a green slot 735 times. Generate a 95% percent confidence around the sample average 0.0735. | ||||||||||||||||||||||||||

108 | Assume we do not know the true value of p=2/37 | ||||||||||||||||||||||||||

109 | (b) Does the true value of p =2/37 lie in this interval? Do you think the roulette wheel is fair? | ||||||||||||||||||||||||||

110 | 18 | March 30 | Tues | Gaussian Processes | https://youtu.be/VAxUactaZeA | ||||||||||||||||||||||

111 | 19 | April 1 | Thurs | Counting/Poisson Processes | https://youtu.be/_PtBsCW5srU | ||||||||||||||||||||||

112 | 1. Cars go by at point on University Avernue at an average rate of 1 car every two seconds. What is the probability no cars go by in one second? | ||||||||||||||||||||||||||

113 | 2. No cars go by for five seconds. What is the probability no cars go by after 6 seconds (i.e. one second more)? | ||||||||||||||||||||||||||

114 | 20 | April 6 | Tues | Stationarity | https://youtu.be/0nYCwtVnOlw | ||||||||||||||||||||||

115 | Homework: | ||||||||||||||||||||||||||

116 | 1. A sequence of iid uniform random numbers on (0,1) is convolved with a rectangular window of length 100. Call the resulting stochastic process X | ||||||||||||||||||||||||||

117 | (a) Calculate the mean, autocorrelation and autocovariance of X | SOLUTION: | https://docs.google.com/document/d/1zPn3xWldlzPtqKMds98j02e01jvNvkDDp8vqK1OV0D4/edit?usp=sharing | ||||||||||||||||||||||||

118 | (b) Is X stationary in the wide sense? | ||||||||||||||||||||||||||

119 | (c) Is X Gaussian or close to Gaussian? | ||||||||||||||||||||||||||

120 | 21 | April 8 | Thurs | Worked Homework | https://youtu.be/1QdpuESFdbg | ||||||||||||||||||||||

121 | 22 | April 13 | Tues | Wide Sense Stationarity | https://youtu.be/azosXf_Pfpk | ||||||||||||||||||||||

122 | Homework: | ||||||||||||||||||||||||||

123 | 1. Explain how you would estimate the autocorrelation of a stationary random process from an ensemble? | ||||||||||||||||||||||||||

124 | 2. Computer work: Do this empicically for the stochastic process defined in Lecture 20 | ||||||||||||||||||||||||||

125 | 23 | April 15 | Thurs | Ergodicity / PSD | https://youtu.be/0xJkv86sewM |