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Exercise 13.7.5. * (Virial Theorem). Since |n, 1, m) is a stationary state, (O)=0 for any...
3. Ehrenfest's Theorem states that dA i for an observable A and a time-independent Hamiltonian H....
3. Ehrenfest's Theorem states that dA i for an observable A and a time-independent Hamiltonian H. Consider a QM system in 1D with time-independent Hamiltonian H 2 V(x). Use Ehrenfest's Theorem to determine Jlx) and p). What is 듦(z)? 4. A projection operator Pn is defined by where there is no summation Ση implied, and the states n> are a complete set of oth onormal states (a) Show that Pa satisfies 2 (b) Show that Pn acting on an arbitrary...
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...
Exercise 1 Consider the regression model through the origin y.-β1zi-ci, where Ei ~ N(0,o). It is ...
Exercise 2b please!
Exercise 1 Consider the regression model through the origin y.-β1zi-ci, where Ei ~ N(0,o). It is assumed that the regression line passes through the origin (0, 0) that for this model a: T N, is an unbiased estimator of o2. a. Show d. Show that (n-D2 ~X2-1, where se is the unbiased estimator of σ2 from question (a). Exercise2 Refer to exercise 1 a. Show that is BLUE (best linear unbiased estimator) b. Show that +1 has...
2. Consider a two state DTMC with state space E = {1 ,2). Let T = min(n > 0 : Xn-1) (i) Compute E...
Equation(1):
2. Consider a two state DTMC with state space E = {1 ,2). Let T = min(n > 0 : Xn-1) (i) Compute E(TIXo = 2) using a geometric distribution. (ii) Use (1) to compute E(T Xo 2). 0), and Vi(n) i rst-step analysis to show that
2. Consider a two state DTMC with state space E = {1 ,2). Let T = min(n > 0 : Xn-1) (i) Compute E(TIXo = 2) using a geometric distribution. (ii) Use...
2 Ehrenfestival Consider a one-dimensional free particle (i.e., subject to no potential, V = 0) of...
2 Ehrenfestival Consider a one-dimensional free particle (i.e., subject to no potential, V = 0) of mass m At a time t = 0, the expectation values of its position and momenta are (2) = zo and (p) = po, respectively. 1. Use Ehrenfest's theorem to find (p(t) 2. Find (t)) using your answer to the previous part and the identification (shown in lecture) that 뚫(z) = m (p).
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that t...
Please all thank you
Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
Exercise 7.1 (Gamblers ruin). Let (Xt) 120 be the Gambler's chain on state space Ω =...
Exercise 7.1 (Gamblers ruin). Let (Xt) 120 be the Gambler's chain on state space Ω = {0, 1,2, , N} (i) Show that any distribution r-[a,0,0, ,0, bl on 2 is stationary with respect to the gambler?s (ii) Clearly the gambler's chain eventually visits state 0 or N, and stays at that boundary state introduced in Example 1.1. chain. Also show that any stationary distribution of this chain should be of this form. thereafter. This is called absorbtion. Let Ti...
r the recurrence relation o. Consider T(n) = Vn T(Vn) + n a. Why can't you solve this with the master theorem? b. S t involves a constant C, tell me what it is in terms of T(O), T(1), or what...
r the recurrence relation o. Consider T(n) = Vn T(Vn) + n a. Why can't you solve this with the master theorem? b. S t involves a constant C, tell me what it is in terms of T(O), T(1), or whatever your inequality by induction. Show the base case. Then show the how that T( n)= 0(n lg ig n). First, clearly indicate the inequality that you wish to hen proceed to prove the inductive hypothesis inductive case, and clearly...
1. To prove the theorem in detail. Theorem: det A for any n X n-matrix A can be computed by a cof...
1. To prove the theorem in detail. Theorem: det A for any n X n-matrix A can be computed by a cofactor expansion across the ith row of A, that is, det A H-1)adtAj Hint: Use induction on i, For the induction step from i to i+1, flip rows i and i+1 (How does this change the determinant?) and use the induction assumption.
1. To prove the theorem in detail. Theorem: det A for any n X n-matrix A can...
Question 8: For any integer n 20 and any real number x with 0
Question 8: For any integer n 20 and any real number x with 0<<1, define the function (Using the ratio test from calculus, it can be shown that this infinite series converges for any fixed integer n.) Determine a closed form expression for Fo(x). (You may use any result that was proven in class.) Let n 21 be an integer and let r be a real number with 0<< 1. Prove that 'n-1(2), n where 1 denotes the derivative of...
3. Ehrenfest's Theorem states that dA i for an observable A and a time-independent Hamiltonian H. Consider a QM system in 1D with time-independent Hamiltonian H 2 V(x). Use Ehrenfest's Theorem to determine Jlx) and p). What is 듦(z)? 4. A projection operator Pn is defined by where there is no summation Ση implied, and the states n> are a complete set of oth onormal states (a) Show that Pa satisfies 2 (b) Show that Pn acting on an arbitrary...
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...
Exercise 2b please!
Exercise 1 Consider the regression model through the origin y.-β1zi-ci, where Ei ~ N(0,o). It is assumed that the regression line passes through the origin (0, 0) that for this model a: T N, is an unbiased estimator of o2. a. Show d. Show that (n-D2 ~X2-1, where se is the unbiased estimator of σ2 from question (a). Exercise2 Refer to exercise 1 a. Show that is BLUE (best linear unbiased estimator) b. Show that +1 has...
Equation(1):
2. Consider a two state DTMC with state space E = {1 ,2). Let T = min(n > 0 : Xn-1) (i) Compute E(TIXo = 2) using a geometric distribution. (ii) Use (1) to compute E(T Xo 2). 0), and Vi(n) i rst-step analysis to show that
2. Consider a two state DTMC with state space E = {1 ,2). Let T = min(n > 0 : Xn-1) (i) Compute E(TIXo = 2) using a geometric distribution. (ii) Use...
2 Ehrenfestival Consider a one-dimensional free particle (i.e., subject to no potential, V = 0) of mass m At a time t = 0, the expectation values of its position and momenta are (2) = zo and (p) = po, respectively. 1. Use Ehrenfest's theorem to find (p(t) 2. Find (t)) using your answer to the previous part and the identification (shown in lecture) that 뚫(z) = m (p).
Please all thank you
Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
Exercise 7.1 (Gamblers ruin). Let (Xt) 120 be the Gambler's chain on state space Ω = {0, 1,2, , N} (i) Show that any distribution r-[a,0,0, ,0, bl on 2 is stationary with respect to the gambler?s (ii) Clearly the gambler's chain eventually visits state 0 or N, and stays at that boundary state introduced in Example 1.1. chain. Also show that any stationary distribution of this chain should be of this form. thereafter. This is called absorbtion. Let Ti...
r the recurrence relation o. Consider T(n) = Vn T(Vn) + n a. Why can't you solve this with the master theorem? b. S t involves a constant C, tell me what it is in terms of T(O), T(1), or whatever your inequality by induction. Show the base case. Then show the how that T( n)= 0(n lg ig n). First, clearly indicate the inequality that you wish to hen proceed to prove the inductive hypothesis inductive case, and clearly...
1. To prove the theorem in detail. Theorem: det A for any n X n-matrix A can be computed by a cofactor expansion across the ith row of A, that is, det A H-1)adtAj Hint: Use induction on i, For the induction step from i to i+1, flip rows i and i+1 (How does this change the determinant?) and use the induction assumption.
1. To prove the theorem in detail. Theorem: det A for any n X n-matrix A can...
Question 8: For any integer n 20 and any real number x with 0<<1, define the function (Using the ratio test from calculus, it can be shown that this infinite series converges for any fixed integer n.) Determine a closed form expression for Fo(x). (You may use any result that was proven in class.) Let n 21 be an integer and let r be a real number with 0<< 1. Prove that 'n-1(2), n where 1 denotes the derivative of...
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