5. If P(AB)-1, show that P(B (i) P(A n B n C) (ii) P(C|A B)-P(CB); and...
(L33*) Verify the following commutation relations (a) [AB,C] A[B,C +IA,CIB (b) [A,|B C]l-IA,BC] [A,CB] The commutator [..] is an important operation in quantum physics. If the elements A.B.C... satisfy the following conditions, they are said to form a Lie-algebra (ab,c are real or complex numbers): ii) [A,Bl -IB.Al iii) The Jacobi identity [AJB CII + [BJCA]] + [CJA,BII :0 - BA Prove the Jacobi identity for [A,B] AB
(L33*) Verify the following commutation relations (a) [AB,C] A[B,C +IA,CIB (b) [A,|B...
(i) Show that 15 (ii) Show that (X) 5/12 and E(Y) 5/8 3(1 - 2X2 +X4) 4(2- 3X +X3) (iii) Show that 3(y|X) (iv) Verify thatE(Y)E(Y) 14] 7. (a) State Chebyshev's inequality and prove it using Markov's inequality 15] (b) Let (2, P) be a probability space representing a random experiment that can be repeated many times under the same conditions, and let A C S2 be a random event. Suppose the experiment is repeated n times (i) Write down...
1. Let X~b(x; n, p) (a) For n 6, p .2, find () Prx> 3), (ii) Pr(x23), (ii) Pr(x (b) For n = 15, p= .8, find (i) Pr(X-2), (ii) Pr(X-12), (iii) Pr(X-8). (c) For n 10, find p so that Pr(X 2 8)6778. く2). 2. Let X be a binomial random variable with μ-6 and σ2-2.4. Fin (a) Pr(X> 2) (b) Pr(2 < X < 8). (c) Pr(Xs 8).
1. Let X~b(x; n, p) (a) For n 6, p...
Prove the following statements. Show your working. (i) If P(A|B,C) = P(B|A,C), then P(A|C) = P(BIC) (ii) If P(A|B,C) = P(A), then P(B,C|A) = P(B,C) (iii) If P(A, B|C) = P(A|C) * P(BIC), then P(A|B,C) = P(AC)
Determine the tension force P. in each of the two cables AB and CB in this system. X A 1 B 4200IB 600 lb X = 24 ft, Y - 10 ft
Choose a generic formula for combination reaction: A) A+B=C B) AB=A+B C) A+BC=AC+B D) AB+CD=AD+CB OB OA
A8.2 Let A be an m × n matrix and B be an n × p matrix. (a) Show that col(B) C null(A) if and only if AB = 0. (b) Show that if AB = 0, then rank(A) + rank(B) 〈 n.
A8.2 Let A be an m × n matrix and B be an n × p matrix. (a) Show that col(B) C null(A) if and only if AB = 0. (b) Show that if AB = 0,...
Let A be an n x p matrix with n p. (a) Show that r(AA) = r(A). (b) Show that I - A(ATA) AT is idempotent. (c) Show that r(1-A(ATAYA") = n-r(A)
Let A be an n x p matrix with n p. (a) Show that r(AA) = r(A). (b) Show that I - A(ATA) AT is idempotent. (c) Show that r(1-A(ATAYA") = n-r(A)
b) X-N(-8,12). Find: i P(X<-9.8) ii. P(X > -8.2) iii. P(-7< X <0.5)
4. Let A and B be n x n such that B = 1-A and A2 = A. Show that AB BA = 0 4. Let A and B be n x n such that B 1-A and A2 = A. Show that AB-BA-0
4. Let A and B be n x n such that B = 1-A and A2 = A. Show that AB BA = 0
4. Let A and B be n x n such that B...