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https://robertmarks.org/Classes/EE5345-Slides/Slides.html
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Sylabus
https://robertmarks.org/Classes/ENGR%205345/5345-Syllabus.pdf
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Text
https://www.amazon.com/gp/product/B00ALTS9TY/
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ECS 5354
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1January 19TuesHistory
Solve the unfinished game problem
https://youtu.be/FMmsinC9q6A
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Due Jan 21
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2January 21ThursReview
Solve the Deal or No Deal Problem
https://youtu.be/-OqhMVIrJOI
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Solve the 5 presidents problem
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Due Jan 28
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3January 28ThursReview
Chapt 2- p89: 77,80 ;p92:104,105
https://youtu.be/m0q7QDDz5-0
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Chapt 4 - p218: 26
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Convert the MatLab erf to my erf. SOLUTION:
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Due Feb 4
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4February 2TuesDistributions
https://youtu.be/9RezsChrzaU
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Info Theory
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5February 4ThursInfo Theory
Evaluate optimal "20 questions" queries for:
https://youtu.be/c7EmERRdq1I
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p = 1/3, 1/3, 1/3 and p = 1/8, 3/8, 1/10, 2/10, 2/10 (hard)
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Evaluate E(Questions) for each and 2^(-length_n) for each
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Compare to the entropy in each case and comment.
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Chapt 3 - p132: 13
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Due Feb 11
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6February 9Tues
RV Transformation
https://youtu.be/hPVmeQri_rc
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7February 11Thurs
Functions of a RV
https://youtu.be/AAG-h8rh5BA
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HOMEWORK:
Show g(x) = F_X transforms a continuous f_X RV to uniform RV
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(a) What if X is a discrete RV? (b) a Posson RV?
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Show inverse of F_X transforms a continuous uniform RV to an f_X RV
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What if X is a discrete RV?
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Find f_Y when X is uniform on (-pi/2,pi/2) and g(x)=tan(x)
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Derive f_Y if X is zero mean Gaussian and (a) Y=|X|, (b) Y=X^2
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Find g(x) to transform a centered Cauchy to a uniform RV
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Due Feb 25
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8February 23Tues
Characteristic Functions
https://youtu.be/skA9oyKv9fg
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HOMEWORK:
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Generate a bunch of Y from a uniform distribution X using a transformation Y=g(X)
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Derive f_Y (y)
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Generate a histogram of Y's, nomalize to unit area and compare to f_Y (y)
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Due March 4
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9February 25Thurs
Characteristic Functions
https://youtu.be/bC9Z9Ls5-bk
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HOMEWORK:
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What is the distribution when the RV's from the following RV's are added:
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SIx geometric RV's all with given parameter p.
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Six Bernoulli RV's all with paramenter p
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Six Pascal Negative Binomial RV's all with parameter p but different r's
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Three gaussians all with different means and variances
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What is the mean and variance when the characteristic function is exp(- |omega| ^N) * exp(-j omega). N is an integer.
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Due March 4
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10March 2Tues2D RV's
https://youtu.be/hWINB89ezfQ
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11March 4Thurs2D RV's
https://youtu.be/UZ68vjyesSc
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HOMEWORK:
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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.
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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
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3. Copy the 10x10 histogram on Sheet 2 of this spreadsheet.
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(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.
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(b) Calculate and generate a 2D plot of the corresponding empirical cumulative distribution function
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(c) Repeat (a) and (b) when we are given that
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(i) 3<=X<=6 and 4<=Y<=6
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(ii) X<Y
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Due March 11
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12March 9Tues2D RV's
https://youtu.be/V4PJUB5oKgc
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13March 11ThursMD RV's
https://youtu.be/KvhONWbthtk
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MIDTERM:
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14March 16TuesMD RV's
https://youtu.be/XyDJnegwzR8
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15March 18Thurs
Law of Large Numbers
https://youtu.be/pj8xdnSBcVw
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Homework
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1. Computer work: Randomly generate points in square. Inside the square is an inscribed circle.
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(a) What is the probability a point will be in the circle?
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(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.
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16March 23Tues
Central Limit Theorem
https://youtu.be/4AvCvR6SJyg
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Homework
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1. Computer work: Generate a bunch of empirical pdf's. For at least one, use a random number generator. Convolve these pdf's.
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(a) Does the convolution have an area of one?
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(b) Plot the convolution and the approximated gaussian in the same figure.
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2. 1000 dollar amounts are rounded UP to the nearest dollar.
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(a) Estimate the total probability the error is greater that \$495. Give a number.
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(b) Greater that \$1000 dollars? Give a number.
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17March 25Thurs
Confidence Intervals
https://youtu.be/oVTWKN5EVII
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1. A roulette wheel has 37 slots. Two of them are green.
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(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.
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Assume we do not know the true value of p=2/37
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(b) Does the true value of p =2/37 lie in this interval? Do you think the roulette wheel is fair?
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18March 30Tues
Gaussian Processes
https://youtu.be/VAxUactaZeA
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19April 1Thurs
Counting/Poisson Processes
https://youtu.be/_PtBsCW5srU
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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?
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2. No cars go by for five seconds. What is the probability no cars go by after 6 seconds (i.e. one second more)?
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20April 6TuesStationarity
https://youtu.be/0nYCwtVnOlw
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Homework:
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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
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(a) Calculate the mean, autocorrelation and autocovariance of X
SOLUTION:
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(b) Is X stationary in the wide sense?
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(c) Is X Gaussian or close to Gaussian?
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21April 8Thurs
Worked Homework
https://youtu.be/1QdpuESFdbg
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22April 13Tues
Wide Sense Stationarity
https://youtu.be/azosXf_Pfpk
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Homework:
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1. Explain how you would estimate the autocorrelation of a stationary random process from an ensemble?
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2. Computer work: Do this empicically for the stochastic process defined in Lecture 20
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23April 15Thurs
Ergodicity / PSD
https://youtu.be/0xJkv86sewM