Verify that Prn (cos θ) solves sin θΟθ (sin ea, Θ) + (E(1 + 1) sin2 θ-m2) Θ 0. Use that pr(z)-(1-z2 )T (4), Pr(r) with Pr(z) a Legendre polynomi 1 Verify that Prn (cos θ) solves sin θΟθ (sin ea, Θ) + (E(1 + 1) sin2 θ-m2) Θ 0. Use that pr(z)-(1-z2 )T (4), Pr(r) with Pr(z) a Legendre polynomi 1
6. (13 marks) where {U, } ~ WN(0,00) is Consider two independent AR(1) series< independent of {K} ~ WN(0,OF). Does their sum Z,-X,-X necessarily follow an AR(1) series? Prove or disprove. (Hint: Compare the causal representation of the sum to that of an AR(1) process) 6. (13 marks) where {U, } ~ WN(0,00) is Consider two independent AR(1) series
6. Consider the system Ar= b given by 2 0 X1 - 1 2 -1 0 - 1 2 X3 Construct the corresponding quadratic P(x1,x2,83), compute its partial derivatives əP/ax, and verify that they vanish exactly at the desired solution.
Find the density function of Y2x+8 9. Let R have probability mass function (pmf) pr)-1/8 for r1,8 Find (I)the cumulative distribution function (cdf) of R; (2)P(R>5): (S)EI(R-3)(R-)) (6)Var(R 10 Suppose the density function of a random variable X is f(x)sige 2- x > 0, where σ>0 is constant. Find E(X) and D()
8. With K(t) = ln(EetX), show that k'(0)-EX], K"(0) = Var(X).
Let {et} denote a white noise process from a normal distribution with E[et] = 0, Var(et) = σe2 and Cov(et, es) = 0 for t ≠ s. Define a new time series {Yt} by Yt = et + 0.6 et -- 1 – 0.4 et – 2 + 0.2 et – 3. 1. Find E(Yt ) and Var(Yt ). 2. Find Cov(Yt , Yt – k) for k = 1, 2, ...
5. [20+5+5] In the regression modely, x,B+ s, pe,+u, ,where I ρ k l and , , let ε, follow an autoregressive (AR) process u' ~ID(Qơ:) , t-l, 2, ,n . <l and u, - Derive the variance-covariance matrix Σ of (q ,6, , , ε" )". From the expression of Σ, identify and interpret Var(.) , t-1, 2, , n . Find the CorrG.ε. and explain its behavior as "s" increases, (s>0). (ii) (iii) 5. [20+5+5] In the regression...
Let(ej denote a white noise process from a normal distribution with E[9] = 0, Var(e-g an Cov(et, e) = 0 for tヂs. Define a new time series {Y.} by Y, = 9 + 0.6 e--04 et-2 + 0.2 9-3 1. Find E(Y) and Var(Y,) 2. Find Cov(Y,X,-k) for k = 1,2,
Below is an atomic instruction: func (var) { var = var + 6; if(var >= 0) { sign = 1; } else { sign = 0; } return sign; } Using func, implement Mutual Exclusion in a multiprocessing system. Also, give the initial value of var and give the entire set of possible initial values for var. Aim to minimize bus traffic.
1. A simple regression model is given by Y81B2X+ e for t 1, (1) ,n errors e with Var (e) a follow AR(1) model where the regression et pet-1 + , t=1...n where 's are uncorrelated random variables with constant variance, that is, E()0, Var (v) = , Cov (, ,) 0 for t Now given that Var (e) = Var (e1-1)= , and Cov (e-1, v)0 (a) Show that (b) Show that E (ee-1)= p. (c) What problem(s) will...