% This is a comment

m=4*5; % this is a matlab command with a comment after it.

Please ask questions if you don't understand any part of the assignment, by email or in class.

1) Matlab coding concepts
Generate a vector x from 0.01 to 1 with increments of 0.01 using colon notation. Now construct y = 5*cos(2*pi*3* x). Use the size() command to determine the size of the vector. Do plot(x,y). Now compute sqrt(2)*std(y). What does this tell you about standard deviations and sinusoids?
Grab the 26 through 75th element of y and put it in a vector w. Create y2 = 5*sin(2*pi*x); Do plot(y,y2); Grab the 1st through 50th element of y2 and put it in a vector w2. Do plot(w,w2). Sort w:

Type

>>help sort

Do:

[w,I]=sort(w); Use the index vector I to sort w2. w2 = w2(I); Now do plot(w2,w). It should look like a zig-zag.

Compute the dot product of y and y2 in four different ways. 1) Using the dot() command. 2) Using a for loop. 3) Using the sum() and .* (the pointwise multiplication operator). 4) Using matrix multiplication * and transpose ' operators.

2) Linear Algebra concepts
% Generate a random x vector.
rx = ceil(10*rand(1,100));
% Generate a random y vector.
ry = ceil(10*rand(1,100));
% do a scatter plot
scatter(rx,ry); axis([0 20 0 20]) % Enter the following matrix
Q = [ 1.2500 0.7500; 0.7500 1.2500]
% Q will transform our x and y vectors. Let's transform all the
% points. We could do it as a for loop:
for j=1:100,
temp = Q*[rx(j); ry(j)]
rxtransform(j) = temp(1);
rytransform(j) = temp(2);
end
scatter(rxtransform,rytransform);
% However, let's do it as a matrix multiply.
% First create a new matrix X by stacking rx and ry to form a 2 by 100 matrix.
% Then compute Y = Q*X. Separate it back into rxtransform and rytransform by
% setting rxtransform and rytransform equal to the 1st and 2nd rows of Y
% respectively
% What kind of transformation does Q perform? (It has a name)
% Every transformation can be decomposed into a rotation U, a scaling S, followed
% by another rotation. Let's use the Singular Value Decomposition to compute
% U and S. [U,S,V]=svd(Q);
% verify that U*S*V' is equal to Q.
% Now U is a rotation matrix, so
% U = [cos(theta) sin(theta); -sin(theta) cos(theta)];
% Use this fact to compute theta (note theta in radians, so multiply result by 180/pi)

3) Probability concepts
% Let's sample from a discrete probability distribution
% Plot the distribution

p = binopdf(1:6,7,0.5);

plot(1:6,p)

% The theoretical expectation is given by: theory_mean = sum(p.*(1:6)); % generate 100 samples from the distribution

r = binornd(7,0.5,100,1);

% compute the empirical expected value of r two ways

% 1) use the mean() function

% 2) generate a uniform weight vector

weight = ones(100,1)/100;

% we can also compute the mean as

emp_mean2 = sum(weight.*r);

% compute the empirical average as weight*r (Why does this work?)

% Let's bin the samples

N = hist(r,1:6);

p_empirical = N/sum(N);

plot(1:6,p_empirical)

% Finally we can compute the empirical mean as

emp_mean3 = sum(p_empirical.*(1:6))

Extra Credit: Simulate the following problems

Birthday problem: What is the probability that at least two people share a birthday in a room with 45 people?

Game Show problem: consider a game show. The host tells you there is a prize behind one of three curtains, A, B, C. You guess A and tell the host. The host then opens curtain B and shows you there is no prize behind that curtain. The host then offers you the opportunity to switch from A to C. Should you switch or stay put or does it matter at all?