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?
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Let Wt de a (Gaussian) white noise with variance σ 2 . Then, let
Xt =...
2. Let (et) be a zero mean white noise process with variance 1. Suppose that the...
2. Let (et) be a zero mean white noise process with variance 1. Suppose that the observed process is h ft + Xt where β is an unknown constant, and Xt-et- Explain why {X.) is stationary. Find its mean function μχ and autocorrelation function p for lk0,1,.. a. b. Show that {Yt3 is not stationary. C. Explain why w. = ▽h = h-K-1 is stationary. d. Calculate Var(Yt) Vt and Var(W) Vt . (Recall: Var(X+c)-Var(X) when c is a constant.)...
2. (a) Consider the following process: where {Z) is a white noise process with unit variance. [1 ...
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...
Let wt for t = . . .,-2,-1, 0, 1, 2, . . . be an...
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
2. Let [et be a zero mean white noise process with variance 0.25. Suppose that the...
2. Let [et be a zero mean white noise process with variance 0.25. Suppose that the observed process is k = et + 0.5e-2. a. Explain why {Yt) is stationary. b. Compute yo-V(Y.) c. Compute the autocorrelation pkY, kl-0,1,2,... for Y) d. Let Wt = 3 + 4t + h. i. Find the mean of {W) ii. Is W3 stationary? Why or why not? iii. Let Z Vw, W,- W,_1. Is {Z.1 stationary? Why or why not?
1. Let {Xt} be a stationary process with mean μt = E(Xt) = 0 and autocovariance...
1. Let {Xt} be a stationary process with mean μt = E(Xt) = 0 and autocovariance function γX(k) = E(XtXt−k) - μ2 = E(XtXt+k) - μ2. De ne Yt = 5 + 2t + Xt. (a) Find E(Yt), the mean function for Yt. (b) Find γY (k), the autocovariance function for Yt in terms of γX (k). (c) Is Yt stationary? Explain. (d) De ne a new process Wt as Wt = Yt − Yt−1. Find E(Wt) and γW (k)....
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...
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...
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?...
For each of the following first construct an example and then show that it has the...
For each of the following first construct an example and then
show that it has the correct properties: (a) (Xt) with constant
mean but has a variance that is a function of time. (b) (Wt) white
noise process that is not strongly stationary. (c) (Zt) is
nonstationary process with an autocovariance function such that
γ(t, t) = σ 2 for all t. (d) (Vt) is nonstationary with an
autocovariance function such that γ(t, t+ h) = 0 for all |h|...
12 Find 10. Let X be a Gaussian rv with mean μ and variance σ, or...
12 Find 10. Let X be a Gaussian rv with mean μ and variance σ, or pdf-l-e 2ơ . Find E X-E(Xt]. Hint: variable substitution, even or odd integrand.
neat writing please 13.7 Let X(t) be a Gaussian white noise with variance o2. It is...
neat writing please
13.7 Let X(t) be a Gaussian white noise with variance o2. It is filtered by a perfect lowpass filter with magnitude HW) = 1 for w<w, and (HW) = 0 for low . What is the autocorrelation function of the filtered signal?
2. Let (et) be a zero mean white noise process with variance 1. Suppose that the observed process is h ft + Xt where β is an unknown constant, and Xt-et- Explain why {X.) is stationary. Find its mean function μχ and autocorrelation function p for lk0,1,.. a. b. Show that {Yt3 is not stationary. C. Explain why w. = ▽h = h-K-1 is stationary. d. Calculate Var(Yt) Vt and Var(W) Vt . (Recall: Var(X+c)-Var(X) when c is a constant.)...
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...
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
2. Let [et be a zero mean white noise process with variance 0.25. Suppose that the observed process is k = et + 0.5e-2. a. Explain why {Yt) is stationary. b. Compute yo-V(Y.) c. Compute the autocorrelation pkY, kl-0,1,2,... for Y) d. Let Wt = 3 + 4t + h. i. Find the mean of {W) ii. Is W3 stationary? Why or why not? iii. Let Z Vw, W,- W,_1. Is {Z.1 stationary? Why or why not?
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?...
For each of the following first construct an example and then
show that it has the correct properties: (a) (Xt) with constant
mean but has a variance that is a function of time. (b) (Wt) white
noise process that is not strongly stationary. (c) (Zt) is
nonstationary process with an autocovariance function such that
γ(t, t) = σ 2 for all t. (d) (Vt) is nonstationary with an
autocovariance function such that γ(t, t+ h) = 0 for all |h|...
12 Find 10. Let X be a Gaussian rv with mean μ and variance σ, or pdf-l-e 2ơ . Find E X-E(Xt]. Hint: variable substitution, even or odd integrand.
neat writing please
13.7 Let X(t) be a Gaussian white noise with variance o2. It is filtered by a perfect lowpass filter with magnitude HW) = 1 for w<w, and (HW) = 0 for low . What is the autocorrelation function of the filtered signal?
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