(2) Recall that a tine seies$43 calusd a white noke procasn (ii) Cov (4,4)=0 Ys#1 ;...
Recall that a time series {εt} is called a white noise process if i. E[εt] = 0 t ; ii. Cov(εs, εt) = 0 s ≠ t ; iii. Var(εt) = σ2 < ∞ Construct the autocorrelation function f(h), h=0,-+1,-+2,… for the white noise process.
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, ...
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,
(b) Let (etez be standard Gaussian, N(0, 1), white noise, and define a first-order au- toregressive conditional homoscedastic, ARCH(1), process by (i) Show that the process {Yt is Markov. (2 marks) (ii) Write down the likelihood for data yi,... . yn from such a process (3 marks) (b) Let (etez be standard Gaussian, N(0, 1), white noise, and define a first-order au- toregressive conditional homoscedastic, ARCH(1), process by (i) Show that the process {Yt is Markov. (2 marks) (ii) Write...
(5 points) Suppose the joint probability mass function (pmf) of integer- Y ī PlX = í,ys j) = (i + 2j)o, for 0 í valued random variables X and < 2,0 < j < 2, and i +j < 3, where c is a constant. In other words, the joint pmf of X and Y can be represented by the table: Y=2 |Y=0 Y=1 X=0| 0 2c 4c 3c 4c 5c X=21 2c (a) Find the constant c. (b) Compute...
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...
Let 9 - {(1,3), (-2,-2)) and 8 = {(-12, 0),(-4,4) be bases for R, and let --12:] be the matrix for T. R2 + R2 relative to B. (2) Find the transition matrix P from 8' to B. P. X (b) Use the matrices P and A to find [v]g and [T()le, where Ivo - [1 -4 [va - [T]8 - I (c) Find p-1 and A' (the matrix for T relative to B). p-1- II A- (d) Find (TV)]g...
Exercise 12: An ASK system employs the following signals in the presence of Additive white noise with a PSD of n/2, t)A c 2f t) for binary 1 So(t)-BA cos(2πfet), for binary 0 where 0< B<1. Derive the probability of error Pe assuming that the binary signals for 1 and 0 occur with equal probability. Hint: Find the average energy per bit Eb Exercise 12: An ASK system employs the following signals in the presence of Additive white noise with...
2. Consider an ARMA(1,1) process, X4 = 0.5X:-1 +0+ - 0.25a4-1, where az is white noise with zero mean and unit variance. (a) Is the model stationary? Explain your answer briefly. (b) Is the model invertible? Explain your answer briefly. (c) Find the infinite moving-average representation of Xt. Namely, find b; such that X =< 0;&–; j=0 (d) Evaluate the first three lags of the ACF and PACF.
2. (a) Consider the following process: where {Z) is a white noise process with unit variance. [1 mark] ii. Find the infinite moving average representation of X,i.e., find the scquence [6 marks] i. Explain why the process is stationary. (6) such that Xt = Σ b,2-j. iii. Calculate the mean and the autocovariance "Yo, γι and 72 of the process. 7 marks iv. Given 40 = 0.1 and Xo = 1.8, find the 2-step ahead forecast of the time series...