[4] 7. Let where X0-0 and Zt comes from WN (0, σ*). Find 7x (s, t)-Cor(X,, X,) for all positive i...
(a) Let X(t) sin t, where is uniformly distributed over the interval [0, 27]. Verify (i) The discrete-time process (t), t = 1,2,... is weakly, but not strongly stationary (ii) The continuous-time stochastic process X(t), t> 0 is neither weakly nor strongly stationary (a) Let X(t) sin t, where is uniformly distributed over the interval [0, 27]. Verify (i) The discrete-time process (t), t = 1,2,... is weakly, but not strongly stationary (ii) The continuous-time stochastic process X(t), t> 0...
Q1) Let X(t) be a zero-mean WSS process with X(t) is input to an LTI system with Let Y(t) be the output. a) Find the mean of Y(t) b) Find the PSD of the output SY(f) c) Find RY(0) ------------------------------------------------------------------------------------------------------------------------- Q2) The random process X(t) is called a white Gaussian noise process if X(t) is a stationary Gaussian random process with zero mean, and flat power spectral density, Let X(t) be a white Gaussian noise process that is input to...
4. Let x E C1 (0,T]; R), T > 0 satisfy _ cos(t)x(t)H(t) + sin(t)2(t) = 0 for all t E (0,T). (a) Show that this defines a FODE for at least one T>0. 1 mark 2 mark (c) Find the potential and conclude (briefly) what is the solution space for the FODE for (b) Transform (possibly inverting) the DE into an exact DE. T. 2 mark 4. Let x E C1 (0,T]; R), T > 0 satisfy _ cos(t)x(t)H(t)...
4. Let (Yi] be a stationary process with mean zero and let a, b and c be constants. Let st be a seasonal with period 4, that is, st-st+4, t-1, 2, . . . , and Xt = a + bt + ct2 + st + Y. (i) Let (ho, do )-min( (k, d)such that k > 0, d 0, and the proces s W t ▽k▽dX,-(1 B)a Find ko and do. For W, (with k = ko and d...
(7) 112 ptsl Let Xi,..., XT denote a random sample of size T from X, where VIX] < oo. a +bX, and for each t define Zt a +bX, for some Define a new random variable Z constants a and b. (a) Show that Z = a + bX and 03-b2q, where the sample me an X and sample variance x of the original sample are as defined in class (b) Prove that Z is an unbiased estimator of E[Z]...
1. Consider the following linear model y Xp+ €, where Cor(e)-021 with ơ e R+ being unknown. an estimable function, where C is a full column rank matrix of rank s. Let T'y be the Let C. β BLUE for CB Write down an explicit expression for T. It should be only in terms of C, y and X. a. basic result do you use to justify your answer? V Cov(Ty). hypothesis is H CB o. (Ty- d), where b....
3. Let U-Bt- tB be Brownian bridge on [0, 1], where {BiJosesi is a Brownian process (i) Show E(Ut0 (ii) Show Cov(U,, Ut) s(1- t) for 0 s ts1. (ii) Let Xg(t)B Find functions g and h such that X, has the same covariance as a Brownian bridge. 3. Let U-Bt- tB be Brownian bridge on [0, 1], where {BiJosesi is a Brownian process (i) Show E(Ut0 (ii) Show Cov(U,, Ut) s(1- t) for 0 s ts1. (ii) Let Xg(t)B...
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...
12. Let x = Vi and y. - 4), where => 0. x b. Find all points(both coordinates) of horizontal tangency. 13. Find the approximate area of one petal of the rose curver - 3cos 30.
|(a) Consider the following function for > 0 f (x)= = -4x 48x (i) Find the stationary point(s) of this function. (3 marks) (ii) Is this function convex or concave? Explain why. (3 marks) (iii What type of stationary point(s) have you found? Include your reasoning. (4 marks) |(b) Show that ln(a) - a has a global maximum and find the value of a > 0 that maximises it. Do the same for ln(a) - a" where n is a...