6.9. A process (Z,, n z is said to be a reverse, or backwards, martingale if EZfor al n and EIZ.I...
6.7. Let X,, be a sequence of independent and identically distributed X, and show Pl random variables with mean 0 and variance σ. Let 1-1 that {Z., n 2 1j is a martingale when 6.7. Let X,, be a sequence of independent and identically distributed X, and show Pl random variables with mean 0 and variance σ. Let 1-1 that {Z., n 2 1j is a martingale when
For a martingale sZ, n 2 1), let X,- Z, - Z,-, i 2 1, where Zo0 Show that Var(Z)-Σ Var(X) 1-1
8.33 Let X1, X2, L(X — м), for n 3D 0, 1, 2, .. (a) Show that Zo, Z\, ... is a martingale with respect to be i.d random variables with mean u < o0. Let Z, = ... Хо. X. 8.33 Let X1, X2, L(X — м), for n 3D 0, 1, 2, .. (a) Show that Zo, Z\, ... is a martingale with respect to be i.d random variables with mean u
Let X,, X,,... be independent and identically distributed (iid) with E X]< co. Let So 0, S,X, n 2 1 The process (S., n 0 is called a random walk process. ΣΧ be a random walk and let λ, i > 0, denote the probability 7.13. Let S," that a ladder height equals i-that is, λ,-Pfirst positive value of S" equals i]. (a) Show that if q, then λ¡ satisfies (b) If P(X = j)-%, j =-2,-1, 0, 1, 2,...
7. X(n) is a zero- discrete-time random process. following input-output relationship: zn) -0.95 mean, stationary, identically and independently, Gaussian distributed white The sample functions of this process is filtered according to the n( zn-1)+x(n) (5 points). Write the MATLAB code for the computation of autocorrelation of the processes X(n) and Z(n) by repeating the experiment 100 times. (5 points). b. 7. X(n) is a zero- discrete-time random process. following input-output relationship: zn) -0.95 mean, stationary, identically and independently, Gaussian distributed...
6. Suppose random variables Z, are exponentially distributed: ZiExp(2) for i 1,2,..., n. Assume that the random variables Z, are independent. For each of the following functions of the Zi, find the expectation E] and variance Var[ ]. (a) 3 Z1- (b) 1.5Z1 +222-3 (c) i 32 (simplify, but final answer is an expression)
QUESTION 3 (a) Consider the ARMA (1, 1) process -Bat-1-where o and θ are model parame- are independent and identically distributed random variables with mean 0 z, oz,-1 ters, and a1, a2, and variance σ (i) Show that the variance of the process is γ,- (ii) Using (i) or otherwise, show that the autocorrelation function (ACF) of the process is: ifk=0. (b) Let Y be an AR(2) process of the special form Y-2Y-2e (i) Find the range of values of...
Suppose that X and Y are independent, identically distributed, geometric random variables with parameter p. Show that P(X = i|X + Y = n) = 1/(n-1) , for i = 1,2,...,n-1
[8 marks] Consider a discrete time stochastic process {Xn,n 2 0j defined by the equation with Xo1 and Rn,n21 are random variables taking their values in (-1,00). Denote Sn-Li-1 Rk for n 〉 1 and So-0 i) [3 marks] Briefly explain why the filtration {F,:n 〉 0} gener- 0 generated by ated by Xo, X1,.. . , Xn and the filtration , n So, S1, , Sn should be identical ії) [5 marks] Show that {X,,n 〉 0} is a...
QUESTION 3 (a) Consider the ARMA(1, 1) process Zt-oZt_itat-θ4-1 :Where φ and θ are model parame- ters, and a, a are independent and identically distributed random variables with mean 0 and variance σ 1-1.4. (i) Show that the variance of the process is γ,- (i) Using () or otherwise, show that the autocorrelation function (ACF) of the process is: if k 0,