25. Consider the LP: min cTx s.t. Ax b, x 0, where A is the 4x2 matrix: [ 0 1; 1 1; 1 2; 1 1], b is a
4x1 vector [5; 9;0; 3] and c is a 2x1 vector [1; 2].
(a) Draw the feasible region in R2.
(b) Draw the contours of cTx =12, cTx =14 and cTx =16 and determ

27. Consider the primal linear program (LP):
min cTx
s.t. ATx b,
x 0 and
its dual:
max bTu
s.t. ATu c
u 0.
Carefully prove that for any feasible (x,u) (i.e. x and u satisfying the constraints of the two LPs), b Tu
cTx.
Solution- We know that
Primal: min

2. Points are sampled uniformly at random from the interval (0,1)2 so that they lie on the line x+y=1.
Determine the expected squared distance between any two sampled points.
Solution: Let A(x1,y1) and B(x2,y2) be two sampled points on line x+y=1 such tha

3. For any two random variables X and Y, the conditional expectation of X given Y=y is defined by
E(X|Y=y)=xpX(x|Y=y) for a fixed y. Show that, for any three random variables A, B and C,
E(A+B|C=c)=E(A|C=c)+E(B|C=c).
Solution: given
E(X|Y=y)=xpX(x|Y=y
To

Thomas Pech
EECS 440
Homework 8
1. Consider a modified SVM formulation derived using the plus-plane at
wx+b=c1 and the minus-plane at wx+b=c2, c1>0, c2<0, c1= c2. Explain the
relationship between the decision surface obtained in this case and the

Thomas Pech
EECS 440
Homework 5
1.i.
For backpropagation to work in a situation where edges are allowed between nodes
in the same layer, you would have to perform a gradient descent that also accounts
for in

Thomas Pech
EECS 440
Homework 9
1. Suppose K1 and K2 are two valid kernels. Show that for positive a and b,
the following are also valid kernels: (i) aK1+bK2 and (ii) aK1K2, where the
product is the Hadamard product: if K=K1K2 then K(x,y)=K1(x,y)K2(x,y).

Thomas Pech
EECS 440
Homework 10
1.
Key:
Blue: Pr(T=1|c1,.,cN)
Orange: Pr(T=2|c1,.,cN)
Yellow: Pr(T=3|c1,.,cN)
yAxis: Pr(T=i|c1,.,cN)
xAxis: number of candies draws from box
(1)
For Box type 1
For Box type 2

Thomas Pech
EECS 440
Homework 7
1. A binary classifier is tested on N independent test sets. The classifier makes
ri errors the ith set, which has size ni. Find the maximum likelihood estimate of
the true error