(White noise is not necessarily i.i.d.). Suppose that {Wt} and {Zt} are independent and identically distributed (i.i.d.) sequences, also independent of each other, with
P(Wt = 0) = P(Wt = 1) = 1/2 and P(Zt = −1) = P(Zt = 1) = 1/2. Define the time series Xt by Xt = . Show that {Xt} is white but not i.i.d.
(White noise is not necessarily i.i.d.). Suppose that {Wt} and {Zt} are independent and identically distributed...
7. Let Z be Gaussian white noise, i.e. Z is a sequence of i.i.d. normal r.v.s each with mean zero and variance 1. Define Zt, t(-1- 1)/v2, if t is odd Show that Xis WN(0,1) (that is, variables Xt and Xt+k,k2 1, are uncorrelated with mean zero and variance 1) but that Xt and Xi-i are not i.i.d 7. Let Z be Gaussian white noise, i.e. Z is a sequence of i.i.d. normal r.v.s each with mean zero and variance...
3. Let Zt) be a Gaussian white noise, that is, a sequence of i.i.d. normal r.v.s each with mean zero and variance 1. Let Y% (a) Using R generate 300 observations of the Gaussian white noise Z. Plot the series and its acf. (b) Using R, plot 300 observations of the series Y -Z. Plot its acf. c) Analyze graphs from (a) and (b). Can you see a difference between the plots of graphs of time series Z and Y?...
Let X1, ..., Xn be i.i.d. [Recall that i.i.d. stands for independent and identically distributed.] Since X1, ..., Xn all have the same distribution, they have the same expected value and variance. Let E(X1) = µ and V ar(X1) = σ 2 . Find the following in terms of µ and σ 2 . (a) E(X2 1 ). Note this is not µ 2 ! (b) E( Pn i=1 X2 i /n). (c) Now, define W by W = 1...
Consider the time series of Xt = Xt−1 + Wt, whereWt are i.i.d and Wt ∼ N (0, σ2 ) and X0 = 0. Let X¯ = 1 n Pn i=1 Xi . Derive the general form for var(X¯). (Hint: Pn i=1 i 2 = n(n+1)(2n+1) 6 ) Consider the time series of X-Xi-1+Wi, whereW are i.i.d and W N(0, σ2) and Xo 0. Let X = n Σ, Derive the general form for var(X). (Hint: Σ i- n(n+1)(2n+1))
Let Wt de a (Gaussian) white noise with variance σ 2 . Then, let Xt = WtWt−1 + µ, where µ is a real constant. Determine the mean and autocovariance of (Xt)? Is this process stationary? Let W, de a (Gaussian) white noise with variance σ2. Then, let of where μ is a real constant. Determine the mean and (X)? Is this process stationary?
Let wt for t = . . .,-2,-1, 0, 1, 2, . . . be an independent and identically distributed process with wt ~ M0, σ2). and consider the time series Determine the mean and the autocovariance function of xt and state whether it is stationary
(a) Suppose that Xi, X2,... are independent and identically distributed random variables each taking the value 1 with probability p and the value-1 with probability 1-p For n 1,2,..., define Yn -X1 + X2+ ...+Xn. Is {Yn) a Markov chain? If so, write down its state space and transition probability matrix. (b) Let Xı, X2, ues on [0,1,2,...) with probabilities pi-P(X5 Yn - min(X1, X2,.. .,Xn). Is {Yn) a Markov chain and transition probability matrix. be independent and identically distributed...
4. Let Xi i = 1,2, ... be independent and identically distributed (i.i.d.) Possion. That is let Px;(k) = Px(k) = 2*24. Let Zn = ??=1 X; and find Pzn. Hint: use characteristic functions. n • Use Li=
Problem 7. Let Xi, X2,..., Xn be i.i.d. (independent and identically distributed) random variables with unknown mean μ and variance σ2. In order to estimate μ and σ from the data we consider the follwing estimates n 1 Show that both these estimates are unbiased. That is, show that E(A)--μ and
(5) Let X1,X2,,Xn be independent identically distributed (i.i.d.) random variables from 1.1 U(0,1). Denote V max{Xi,..., Xn) and W min{Xi,..., Xn] (a) Find the distributions and the densities and the distributions of each of V and W. (b) Find E(V) and E(W) (5) Let X1,X2,,Xn be independent identically distributed (i.i.d.) random variables from 1.1 U(0,1). Denote V max{Xi,..., Xn) and W min{Xi,..., Xn] (a) Find the distributions and the densities and the distributions of each of V and W. (b)...