Homework Answers
4. [10 pts] Define T: P2 → Max2 by T(p)=Ip(0) p(-1) 10(1) p(2) (a) Show that...
4. [10 pts] Define T: P2 → Max2 by T(p)=Ip(0) p(-1) 10(1) p(2) (a) Show that T is a linear transformation. (b) What is the kernel of T? (e) Describe the range of T as the set of 2 x 2 matrices with the entries a,b.c,d satisfying certain conditions. Hint: Work on the practice problem §4.231 first. For finding the kernel, you may use the fact that if a polynomial p has degree at most 3 and p(0) = p(-1)...
Consider a DTMC X;n 2 0 with state space E 0,1,2,... ,N), and transition probability matrix P = (...
Consider a DTMC X;n 2 0 with state space E 0,1,2,... ,N), and transition probability matrix P = (pij). Define T = min(n > 0 : Xn-0), and vi(n) = P(T > n|X0 = i). Use the first-step analysis to show that vi (72), t"2(n), . . . , UN(n)) = where B is a submatrix of P obtained by deleting the row and column corresponding to the state 0. Hint: First establish a recursive formula v(n )-ΣΝ1pijuj(n-1).
Consider a...
Define four sets of integers Let P {0, 1), let Q {-11, 1, 5) , and...
Define four sets of integers Let P {0, 1), let Q {-11, 1, 5) , and Let R and S be arbitrary nonempty subsets of Z. Define an even indicator function F F: ZP by F(x) = (x + 1) mod 2 for x e Z That is, F(x) 1 if x is even, and F(x) = 0 if x is odd. or neither? Explain. a) Is F: Q P one-to-one, onto, both, or neither? Explain. b) Is F: (Pn...
4. For this question, we define the following matrices: 1-2 0 To 61 C= 0 -1...
4. For this question, we define the following matrices: 1-2 0 To 61 C= 0 -1 2 , D= 3 1 . [3 24 L-2 -1] (a) For each of the following, state whether or not the expression can be evaluated. If it can be, evaluate it. If it cannot be, explain why. i. B? +D ii. AD iii. C + DB iv. CT-C (b) Find three distinct vectors X1, X2, X3 such that Bx; = 0 for i =...
xerc 0 e:0 y ln(r), Let Dbe the following two dimensional donain. D := {(zy) E...
xerc 0 e:0 y ln(r), Let Dbe the following two dimensional donain. D := {(zy) E R2 : 1 S$ where In denotes the natural logarithm and the umber e denotes its base, ie. e 2.71. (1) Sketch the domain D and compute its area. (2) Let us define the domain D as the part of the rectangle [I.ej x (o, I] that is above D. Sketch D and compute its area. (3) Compute Now rotate the domain D around...
(5) Fixm 2 1, an integer, and suppose P~ Uniform([0, 1]) and N ~Binomial(m, P) (a) Determine E(Xk...
(5) Fixm 2 1, an integer, and suppose P~ Uniform([0, 1]) and N ~Binomial(m, P) (a) Determine E(Xk(NP) where χκ (n), k-0, 1, 2, . . . , are defined as follows: 1 if n-k 0 otherwise (b) Determine E(Xk(N)h(N)) for a general function h : R R (c) Determine E(PIN) Warning: E(PN) is not N/m as you might be tempted to guess. Hint: Use the law of total probability together with the following result which you showed (in greater...
4. For this question, define f(x) = (x - 1)e-(0-1). (a) Find f'(x) and f"(x). (b)...
4. For this question, define f(x) = (x - 1)e-(0-1). (a) Find f'(x) and f"(x). (b) Find where S is increasing and where / is decreasing (e) Find where S is concave up and where / is concave down. (a) Find all critical points of . For each point you find, explain whether it is a (relative) maximum, a (relative) minimum or neither. (e) Find all points of inflection of f. For each point you find, explain why it is...
The helicity operator is defined as h = sigma middot p/|p| where sigma = (sigma 0 0 sigma). Lets also define positive an...
The helicity operator is defined as h = sigma middot p/|p| where sigma = (sigma 0 0 sigma). Lets also define positive and negative helicity projection operators P^h_plusminus = (1 plusminus h)/2. Use natural units, h = c = 1. (a) By using the Dirac equation for a fermions with spinor u(p, s), mass m and energy E, show that P^h_plusminus = 1/2 (1 + gamma^5_|P| (E - beta m)). (b) For massless spin-1/2 particles like neutrino, show that the...
3. [6 marks] Diagonalization: IfAs | 2-2 | (as in Q1), and P= 11-1 1 -1 0 1 11-1 (a) Verify (by calculating P-1p) t...
3. [6 marks] Diagonalization: IfAs | 2-2 | (as in Q1), and P= 11-1 1 -1 0 1 11-1 (a) Verify (by calculating P-1p) that P10 1 is the inverse matrix of P. ents only on the leading diagonal. (c) Compare the eigenvalues of A (from Ql) to D and the eigenvectors of A to the columns of P. What do you notice?
3. [6 marks] Diagonalization: IfAs | 2-2 | (as in Q1), and P= 11-1 1 -1 0...
Please answer d,e,f and g, thank you! roblem 1. Let (U common p.d.f. i 1 be...
Please answer d,e,f and g, thank you!
roblem 1. Let (U common p.d.f. i 1 be a sequence of ii.d. discrete random variables with f(k) for k = 1, 2, 3 and for n 21 let Sn = Σ,u. (a) Find the probability that S2 is even. (b) Find the probability that Sn is even given that S,-1 is even. (e) Find the probability that Sn is even given that S-1 is odd. (d) Let pn P(Sn is even). Find...
4. [10 pts] Define T: P2 → Max2 by T(p)=Ip(0) p(-1) 10(1) p(2) (a) Show that T is a linear transformation. (b) What is the kernel of T? (e) Describe the range of T as the set of 2 x 2 matrices with the entries a,b.c,d satisfying certain conditions. Hint: Work on the practice problem §4.231 first. For finding the kernel, you may use the fact that if a polynomial p has degree at most 3 and p(0) = p(-1)...
Consider a DTMC X;n 2 0 with state space E 0,1,2,... ,N), and transition probability matrix P = (pij). Define T = min(n > 0 : Xn-0), and vi(n) = P(T > n|X0 = i). Use the first-step analysis to show that vi (72), t"2(n), . . . , UN(n)) = where B is a submatrix of P obtained by deleting the row and column corresponding to the state 0. Hint: First establish a recursive formula v(n )-ΣΝ1pijuj(n-1).
Consider a...
Define four sets of integers Let P {0, 1), let Q {-11, 1, 5) , and Let R and S be arbitrary nonempty subsets of Z. Define an even indicator function F F: ZP by F(x) = (x + 1) mod 2 for x e Z That is, F(x) 1 if x is even, and F(x) = 0 if x is odd. or neither? Explain. a) Is F: Q P one-to-one, onto, both, or neither? Explain. b) Is F: (Pn...
4. For this question, we define the following matrices: 1-2 0 To 61 C= 0 -1 2 , D= 3 1 . [3 24 L-2 -1] (a) For each of the following, state whether or not the expression can be evaluated. If it can be, evaluate it. If it cannot be, explain why. i. B? +D ii. AD iii. C + DB iv. CT-C (b) Find three distinct vectors X1, X2, X3 such that Bx; = 0 for i =...
xerc 0 e:0 y ln(r), Let Dbe the following two dimensional donain. D := {(zy) E R2 : 1 S$ where In denotes the natural logarithm and the umber e denotes its base, ie. e 2.71. (1) Sketch the domain D and compute its area. (2) Let us define the domain D as the part of the rectangle [I.ej x (o, I] that is above D. Sketch D and compute its area. (3) Compute Now rotate the domain D around...
(5) Fixm 2 1, an integer, and suppose P~ Uniform([0, 1]) and N ~Binomial(m, P) (a) Determine E(Xk(NP) where χκ (n), k-0, 1, 2, . . . , are defined as follows: 1 if n-k 0 otherwise (b) Determine E(Xk(N)h(N)) for a general function h : R R (c) Determine E(PIN) Warning: E(PN) is not N/m as you might be tempted to guess. Hint: Use the law of total probability together with the following result which you showed (in greater...
4. For this question, define f(x) = (x - 1)e-(0-1). (a) Find f'(x) and f"(x). (b) Find where S is increasing and where / is decreasing (e) Find where S is concave up and where / is concave down. (a) Find all critical points of . For each point you find, explain whether it is a (relative) maximum, a (relative) minimum or neither. (e) Find all points of inflection of f. For each point you find, explain why it is...
3. [6 marks] Diagonalization: IfAs | 2-2 | (as in Q1), and P= 11-1 1 -1 0 1 11-1 (a) Verify (by calculating P-1p) that P10 1 is the inverse matrix of P. ents only on the leading diagonal. (c) Compare the eigenvalues of A (from Ql) to D and the eigenvectors of A to the columns of P. What do you notice?
3. [6 marks] Diagonalization: IfAs | 2-2 | (as in Q1), and P= 11-1 1 -1 0...
Please answer d,e,f and g, thank you!
roblem 1. Let (U common p.d.f. i 1 be a sequence of ii.d. discrete random variables with f(k) for k = 1, 2, 3 and for n 21 let Sn = Σ,u. (a) Find the probability that S2 is even. (b) Find the probability that Sn is even given that S,-1 is even. (e) Find the probability that Sn is even given that S-1 is odd. (d) Let pn P(Sn is even). Find...
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