Assignment #1 Let R be the space of all the N! permutations of (1,2,...,N), and let...

 

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Assignment #1 Let R be the space of all the N! permutations of (1,2,...,N), and let R = (RR) be uniformly distributed over R. 1. Prove that (1) (11) (111) P (R₂-k); 11, kN. N 2. (1) P(RkRh)(N-1) 1<11k+h< P(R₁-k, R₁-k) -0, 1<1+1,k<N. B[a(R₂)] - (11) V[a(R₂)]- (a,-5) ²/N j-1 N N (111) Cov[ª(R₂),ª(R;)] = - (²-1) 21 (4-3) ² k-1 > N where a (1) = a and a=1 2aji N 1-1 N 3. Set S = ₁a (R₁), where the c's are constants. i=1 Then N (1) E(S) = Na where - 1,₁ N 1-j (a₂-a) ²₁ (₁-c) ² N (11) V(S) = (N-15¹ E 1-1 R 4. Set S = I a(R₂), N and let R (R₁,...,R) be uniformly distributed over R. 1-1 Then prove that (1) E(S) ==ā (11) V(S) = n(N-m) N N(N-1) 1-1 (4-0) ². m 5. Let W₂ = ER; i=1 n n = EST. W₁ = Assignment #1 Let R be the space of all the N! permutations of (1,2,...,N), and let R = (RR) be uniformly distributed over R. 1. Prove that (1) (11) (111) P (R₂-k); 11, kN. N 2. (1) P(RkRh)(N-1) 1<11k+h< P(R₁-k, R₁-k) -0, 1<1+1,k<N. B[a(R₂)] - (11) V[a(R₂)]- (a,-5) ²/N j-1 N N (111) Cov[ª(R₂),ª(R;)] = - (²-1) 21 (4-3) ² k-1 > N where a (1) = a and a=1 2aji N 1-1 N 3. Set S = ₁a (R₁), where the c's are constants. i=1 Then N (1) E(S) = Na where - 1,₁ N 1-j (a₂-a) ²₁ (₁-c) ² N (11) V(S) = (N-15¹ E 1-1 R 4. Set S = I a(R₂), N and let R (R₁,...,R) be uniformly distributed over R. 1-1 Then prove that (1) E(S) ==ā (11) V(S) = n(N-m) N N(N-1) 1-1 (4-0) ². m 5. Let W₂ = ER; i=1 n n = EST. W₁ =

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