Solve the unfinished game problem
|Due Jan 21|
Solve the Deal or No Deal Problem
Solve the 5 presidents problem
|Due Jan 28|
Chapt 2- p89: 77,80 ;p92:104,105
|Chapt 4 - p218: 26|
Convert the MatLab erf to my erf. SOLUTION:
|Due Feb 4|
|5||February 4||Thurs||Info Theory|
Evaluate optimal "20 questions" queries for:
p = 1/3, 1/3, 1/3 and p = 1/8, 3/8, 1/10, 2/10, 2/10 (hard)
Evaluate E(Questions) for each and 2^(-length_n) for each
Compare to the entropy in each case and comment.
|Chapt 3 - p132: 13|
|Due Feb 11|
Functions of a RV
Show g(x) = F_X transforms a continuous f_X RV to uniform RV
(a) What if X is a discrete RV? (b) a Posson RV?
Show inverse of F_X transforms a continuous uniform RV to an f_X RV
What if X is a discrete RV?
Find f_Y when X is uniform on (-pi/2,pi/2) and g(x)=tan(x)
Derive f_Y if X is zero mean Gaussian and (a) Y=|X|, (b) Y=X^2
Find g(x) to transform a centered Cauchy to a uniform RV
|Due Feb 25|
Generate a bunch of Y from a uniform distribution X using a transformation Y=g(X)
|Derive f_Y (y)|
Generate a histogram of Y's, nomalize to unit area and compare to f_Y (y)
|Due March 4|
What is the distribution when the RV's from the following RV's are added:
SIx geometric RV's all with given parameter p.
Six Bernoulli RV's all with paramenter p
Six Pascal Negative Binomial RV's all with parameter p but different r's
Three gaussians all with different means and variances
What is the mean and variance when the characteristic function is exp(- |omega| ^N) * exp(-j omega). N is an integer.
|Due March 4|
|10||March 2||Tues||2D RV's|
|11||March 4||Thurs||2D RV's|
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.
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
3. Copy the 10x10 histogram on Sheet 2 of this spreadsheet.
(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.
(b) Calculate and generate a 2D plot of the corresponding empirical cumulative distribution function
(c) Repeat (a) and (b) when we are given that
(i) 3<=X<=6 and 4<=Y<=6
|Due March 11|
|12||March 9||Tues||2D RV's|
|13||March 11||Thurs||MD RV's|
|14||March 16||Tues||MD RV's|
Law of Large Numbers
1. Computer work: Randomly generate points in square. Inside the square is an inscribed circle.
(a) What is the probability a point will be in the circle?
(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.
(c) Does the limit of your plot approach your answer in (a)?
Central Limit Theorem
1. Computer work: Generate a bunch of empirical pdf's. For at least one, use a random number generator. Convolve these pdf's.
(a) Does the convolution have an area of one?
(b) Plot the convolution and the approximated gaussian in the same figure.
2. 1000 dollar amounts are rounded UP to the nearest dollar.
(a) Estimate the total probability the error is greater that $495. Give a number.
(b) Greater that $1000 dollars? Give a number.
1. A roulette wheel has 37 slots. Two of them are green.
(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.
Assume we do not know the true value of p=2/37
(b) Does the true value of p =2/37 lie in this interval? Do you think the roulette wheel is fair?
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?
2. No cars go by for five seconds. What is the probability no cars go by after 6 seconds (i.e. one second more)?
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
(a) Calculate the mean, autocorrelation and autocovariance of X
(b) Is X stationary in the wide sense?
(c) Is X Gaussian or close to Gaussian?
Wide Sense Stationarity
1. Explain how you would estimate the autocorrelation of a stationary random process from an ensemble?
2. Computer work: Do this empicically for the stochastic process defined in Lecture 20
Ergodicity / PSD