Question 20: Quantum Probability and Spectral Measures Problem Statement: In quantum probability, observables are represented by self-adjoint operators on a Hilbert space H. Let X be a self-adjoint operator representing a quantum observable with spectral measure E. 1. Probability Measures from Quantum States: Given a quantum state p (a positive trace-class operator with trace one), define the probability measure μp on R by μ(B) = Tr(pE(B)) for Borel sets B. Prove that up is a well-defined probability measure. 2. Measure-Theoretic Expectation and Variance: Show that the expectation E(X) and variance Var, (X) of the observable X in the state p can be expressed as E(X) = [Adµ(A), Var(x) = [ {A – E‚(X))² dµ‚(A). Provide detailed proofs of these expressions using measure-theoretic integration. 3. Spectral Measures and Quantum Measurements: Analyze how the spectral measure E encodes the probabilities of different measurement outcomes for X and prove that successive measurements correspond to measure-theoretic operations on the spectral measures associated with X. Requirements: Connect spectral theory with the foundations of quantum probability.

(Revision.) Let X = Wo.5 + 0.5W1 – 2W2 – W3, where (W1, t > 0) is standard BM. Find the expectation E(X²).

Question 5 The discrete random variable X has probability function f(x) given by f(x) = k (lxl + 1)²,x = -1,0,1 a) Find the value of k. b) Find the following values: ) E(X) ii) E(X²) ilii) E (3X² – 2X + 4) iv) Var(X) v) Var(3X – 7) 2.

(a+b) (a+0+1) Derive an incremental update formula to estimate Question 3. the covariance matrix. In other words given {x1,..., xN} how do you update the ML estimate of the covariance matrix EN for xN+1 to get EN+1•

Example 8-20. how that : E(2) = 10-1 G(t) dt, a > Ó - (* X+ muhere G(t) is the probability generating function of X. Find it when X ~ B (, p), and a = 1.

P8

3. FIND THE PROBABILITY P(Z > -1.85 ) 4 In figure b-2, if the 2 place % large area (probability) = 77.2%, how much area is in the right tail ? Find 2 z scores, one positive and one negative, that are equidistant from the mean so that the areas in the two tạils total 12%. If a z score is found 4 place decimal 6. Like 0.7563 to be 0.62, what is the resulting probability (AREA) TO THE LEFT OF THE Z S CORE

question (a) to (d)

Conditional entropy. Let A and B = (Bo, B₁,..., Bn) be a discrete random variable and vector, respectively. The conditional entropy of A with respect to B is defined as H(A | B) = E(E{-log f(A | B) | B}) where f(a | b) = P(A = a | B = b). Let X be an aperiodic Markov chain on a finite state space. Show that H(Xn+1 | X0, X₁. , Xn) = H(Xn+1 | Xn), and that H(Xn+1 \ Xn) → - ΣπιΣ pi; i j if X is aperiodic with a unique stationary distribution . log Pij as n →∞,

Consider a probability space (2, F, P) and consider a random variable, X Let Fx (x) be the distribution function of X Prove that Fx (t) - limno Fx (t-n-¹)=P{ X=t}.