課程名稱︰多變量統計分析 課程性質︰應數所數統組必修 課程教師︰陳宏 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2016/4/26 考試時限（分鐘）：15:30~17:20 試題 : 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]< ---. c (b) When we pick c=exp(0.01p) random vectors as above, what can you say about whether the vectors is almost orthogonal? 3. (25%) Consider the following matrix _ _ | | | 1 a | A = | | ε | ε 1 | |_ _| where a>0. Describe the change of the eigenvalues of A when ε moves away ε from 0. 4. (30%) Using the matrix ( 4 8 8 ) A = ( ) ( 3 6 9 ) T (a) Calculate AA and obtain its eigenvalues and eigenvectors. T (b) Calculate A A and obtain its eigenvalues and eigenvectors. Check that the nonzero eigenalues are the same as those in part(a). (c) Obtain the singular value decomposition of A. 5. (25%) (X ,X ) is following a bivariate normal distribution with density 1 2 function f(x ,x )= 1 2 2 2 (x -μ) x -μ x -μ (x -μ) 1 1 1 1 1 1 2 2 2 2 —————————exp{-—————[————-2ρ——— ———+————]} 2 2 2 2 2πσσ√(1-ρ ) 2(1-ρ ) σ σ σ σ 1 2 1 1 2 2 Determine the major axis of the ellipsoid corresponding to a constant joint density, i.e. f(x ,x )=c, where c is a real positive constant. 1 2 T 6. (40%) Let V be a p×p covarince matrix of a random vector x=(X ,...,X ) . 1 p T Consider a linear combination a x with ||a||=1. T (a) Find a which maximizes a Va over ||a||=1. 1 T T (b) Derive a which maximizes a Va over ||a||=1 and a a = 0. 2 2 1 7. (40%) Let V be a p×p covarince matrix of a random vector x=(X ,...,X ) . 1 p T p Consider a linear combination a x with Σ a = 1 and Σ a μ=μ. i i i=1 i i 0 T (a) Find a which maximizes a Va with the just-mentioned constraints. 1 T T (b) Derive a which maximizes a Va with the same constraints and a a = 0. 2 2 1 8. (30%) Let r be the return of a stock index at time t. Consider Sharpe's 0t single-index model. It assumes that the log returns of the n stocks in the index are generated by r = α+βr +ε . it i i 0t it 2 Here α and β are constants,ε is uncorrelated with r , Var(ε )=σ , and i i it 0t it Cov(ε ,ε )=0. Often, r is the return of a market index. We further it jt 0t assume that (r ,...,r ), 1≦t≦n, are i.i.d. random vector. Suppose 0t pt 2 Var(r )=σ . Please derive the covariance of r . 0t 0 it 9. (40%) (Hidden Markov Model) Consider a coin flipping experiment in which we have coin 1 with probability of getting head 0.51 while coin 2 has probability of getting head 0.49. Assume that coin 1 will be picked with probability 0.01. The transition porbabilities from state 1 to state 1 and state 2 are 0.9 and 0.1, respectively while the transition probabilities from state 2 to state 1 and state 2 are 0.1 and 0.9, respectively. (a) Now, suppose we observe three coin flips all resulting in heads. The sequence of observations is therefore (H,H,H). What is the most likely state sequence given these three observations? (b) What happens to the most likely state sequence if we observe a long 6 sequence of all heads (e.g., 10 heads in a row)? 10. (30%) When P(A|B∩C)=P(A|C), show that P(A∩B|C)=P(A|C)P(B|C). --

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