课程名称︰多变量统计分析
课程性质︰应数所数统组必修
课程教师︰陈宏
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2016/4/26
考试时限（分钟）：15:30~17:20
试题 :
1. (25%)In class, we talk about the norm of the random projection approximately
maintained. By knowing that
2 2 2 2 3
P((1-ε)||x|| ≦||ψ(x)|| ≦(1+ε)||x|| )≧1-2exp(-(ε-ε)m/4),
p
where ε∈(0,1/2). For any u,v∈R with ||u||≦1 and ||v||≦1, give a
T T
probability upper bound on the following event holds {|u v-ψ(u) ψ(v)|>ε}.
2. (40%) If X and Y are chosen at random from -1 and 1 with probabilities 1/2,
i i
T T
respectively. Let X=(X ,...,X ) and Y=(Y ,...,Y ) . Denote the angle between
1 p 1 p
Y and X by θ.
1/2 1
(a) Show that P[|cos(θ)|>(log c) /√p]<