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Let { be a zero-mean stationary process and let a and b be constants. (a) (5...
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,...
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...
I. (5 points) Let {X, } be a stationary series with mean μ and autocovariance function...
I. (5 points) Let {X, } be a stationary series with mean μ and autocovariance function 7(), and icz Show Y is also stationary for a, ER, iE Z 2. (5 points) Let {Xi be the process Xi A cos(wt) Bsin(t),t 1,2, ., COS
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)....
Yt = 5 − 2t + Xt, where {Xt} is stationary with mean 0 and autocovariance...
Yt = 5 − 2t + Xt, where {Xt} is stationary with mean 0 and autocovariance function γk. Now, let Wt = Yt − Yt−1. (a) Find the mean function for {Wt}. (b) Find the autocovariance function for {Wt}. (c) Is {Wt} stationary? Why or why not?
3. Let X1, . . . , Xn be iid random variables with mean μ and...
3. Let X1, . . . , Xn be iid random variables with mean μ and variance σ2. Let X denote the sample mean and V-Σ,(X,-X)2 a) Derive the expected values of X and V b) Further suppose that Xi,...,Xn are normally distributed. Let Anxn - ((a) be an orthogonal matrix whose first row is (mVm Y = (y, . . . ,%), and X = (Xi, , Xn), are (column) vectors. (It is not necessary to know aij for...
5. A stationary random process X[n] is input to a discrete time LTI system with frequency...
5. A stationary random process X[n] is input to a discrete time LTI system with frequency response j“)-10 zero mean given as A(e nmay be expressed as where Wnlis a zero mea a-HS1 unit variancei.i.d. (independent identically distributed) Gaussian sequence and c, d are constants. Let Yl be the output random a)Determine the mean function for the output random sequence Yn in terms ofa, c and d b) Determine S7 (e), the power spectral density ofthe output random sequence Yn]...
Please ignore part abc 4. Suppose that (X1, Yİ), , (XN,Yv) denotes a random sample. Let Si = a + bX, T, e+ dy, where a, b, c and d are constants. Let X ΣΧ, and σ2-NL Σ(x,-x)2, with the analogous expr...
Please ignore part abc
4. Suppose that (X1, Yİ), , (XN,Yv) denotes a random sample. Let Si = a + bX, T, e+ dy, where a, b, c and d are constants. Let X ΣΧ, and σ2-NL Σ(x,-x)2, with the analogous expressions for y S, T. Let σΧΥ-ΝΤΣ (Xi-X)(X-Y), and let P:XY ƠXY/(ƠXƠY), with the analogous expressions for S, T. (a) Show that σ bbe (b) Show that ớsı, d ớx (c) Show that psT ST (d) How do the...
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.)...
3. Let Xi, . . . , Xn be iid randoln variables with mean μ and...
3. Let Xi, . . . , Xn be iid randoln variables with mean μ and variance σ2. Let, X denote the sample mean and V-Σ, (X,-X)2. (a) Derive the expected values of X and V. (b) Further suppose that Xi,-.,X, are normally distributed. Let Anxn ((a)) an orthogonal matrix whose first rOw 1S be , ..*) and iet Y = AX, where Y (Yİ, ,%), ard X-(XI, , X.), are (column) vectors. (It is not necessary to know aij...
5. (4 points) Let X1, X2, be independent random variables that are uniformly distributed on [-1,1]...
5. (4 points) Let X1, X2, be independent random variables that are uniformly distributed on [-1,1] Show that the sequence Yi,Y2, converges in probability to some limit, and identity the limit, for each of the following cases: (a) Yn = max Xi, , xn (this is similar to an example from class). (c) Yn = (Xn)"
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...
I. (5 points) Let {X, } be a stationary series with mean μ and autocovariance function 7(), and icz Show Y is also stationary for a, ER, iE Z 2. (5 points) Let {Xi be the process Xi A cos(wt) Bsin(t),t 1,2, ., COS
3. Let X1, . . . , Xn be iid random variables with mean μ and variance σ2. Let X denote the sample mean and V-Σ,(X,-X)2 a) Derive the expected values of X and V b) Further suppose that Xi,...,Xn are normally distributed. Let Anxn - ((a) be an orthogonal matrix whose first row is (mVm Y = (y, . . . ,%), and X = (Xi, , Xn), are (column) vectors. (It is not necessary to know aij for...
5. A stationary random process X[n] is input to a discrete time LTI system with frequency response j“)-10 zero mean given as A(e nmay be expressed as where Wnlis a zero mea a-HS1 unit variancei.i.d. (independent identically distributed) Gaussian sequence and c, d are constants. Let Yl be the output random a)Determine the mean function for the output random sequence Yn in terms ofa, c and d b) Determine S7 (e), the power spectral density ofthe output random sequence Yn]...
Please ignore part abc
4. Suppose that (X1, Yİ), , (XN,Yv) denotes a random sample. Let Si = a + bX, T, e+ dy, where a, b, c and d are constants. Let X ΣΧ, and σ2-NL Σ(x,-x)2, with the analogous expressions for y S, T. Let σΧΥ-ΝΤΣ (Xi-X)(X-Y), and let P:XY ƠXY/(ƠXƠY), with the analogous expressions for S, T. (a) Show that σ bbe (b) Show that ớsı, d ớx (c) Show that psT ST (d) How do 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.)...
3. Let Xi, . . . , Xn be iid randoln variables with mean μ and variance σ2. Let, X denote the sample mean and V-Σ, (X,-X)2. (a) Derive the expected values of X and V. (b) Further suppose that Xi,-.,X, are normally distributed. Let Anxn ((a)) an orthogonal matrix whose first rOw 1S be , ..*) and iet Y = AX, where Y (Yİ, ,%), ard X-(XI, , X.), are (column) vectors. (It is not necessary to know aij...
5. (4 points) Let X1, X2, be independent random variables that are uniformly distributed on [-1,1] Show that the sequence Yi,Y2, converges in probability to some limit, and identity the limit, for each of the following cases: (a) Yn = max Xi, , xn (this is similar to an example from class). (c) Yn = (Xn)"
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