Exercise 1 (Self-learning)
Log into “cookdata.cn”, and enroll the course “êâ‰Æ Ú”. The online homework is given there. The system will judge your answers.
Exercise 2 (Written Problem)
Given the ordered data {x(i )}2i˘n1¡1 data set is equal to the minimizer
with increasing order. Show that the median of the of the following L1 minimization problem:
2n¡1
min
X
x(n) ˘ arg
jx(i ) ¡cj.
c
i ˘1
Exercise 3 (Written Problem)
Consider the probability density function (PDF) shown in the following figure and equa-
tions:
p(x)
8
22x
if x
˙
0,
˘
>
0,
w ¡ w2 ,
if 0 É x É w,
>
<
>
0,
ifw ˙ x.
>
:
1. Which of the following expression is true? (Only one truth.)
(A) E[X ] ˘ R¡11( w2 ¡ w2x2 )dx;
(B) E[X ] ˘ R¡11 x( w2 ¡ w2x2 )dx;
1
(C) E[X ] ˘ R¡11 w( w2 ¡ w2x2 )dx;
(D) E[X ] ˘ R0w ( w2 ¡ w2x2 )dx;
(E) E[X ] ˘ R0w x( w2 ¡ w2x2 )dx;
(F) E[X ] ˘ R0w w( w2 ¡ w2x2 )dx;
2. What is P(x ˘ 1jw ˘ 2)?
3. When w ˘ 2, what is p(1)? Exercise 4 (Written Problem)
Let X and Y be two continuous random variables. The conditional expectation of Y on X ˘ x is defined as the expectation of Y with respect to the conditional probability density p(Y jX ):
E(Y jX ˘ x) ˘ ZY
y p(yjX ˘ x)dy ˘
RY ypx (x)
,
p(x, y)dy
where px (x) is the marginal probability density of Y . Show the following properties of the conditional expectation:
1. Epy Y ˘ Epx [E(Y jX )], where Epy means taking the expectation with respect to the marginal probability density py .
Remark: This formula is sometimes called the tower rule.
2. If X and Y are independent, then E(Y jX ˘ x) ˘ E(Y ). Exercise 5 (Written Problem)
T
The Jaccard distance between two sets A and B is defined as J–(A, B) ˘ 1 ¡ jjAA BBjj ˘
jA4Bj
Jaccard
, where jSj stands for the number of elements in the set S. Show that the
S
jA
S
Bj
–
is actually a distance, i.e., it satisfies the three properties:
distance J
1. Positivity: J–(A, B) ˚ 0, and “=” if and only if A ˘ B;
2. Symmetry: J–(A, B) ˘ J–(B, A);
3. Triangle inequality: J–(A, B) É J–(A,C ) ¯ J–(B,C ).
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