Suppose that Z1 and Z2 are uncorrelated random variables with
zero mean and unit variance. Consider the process defined by
Yt = Z1 cos(ωt) + Z2 sin(ωt) + et
where et ∼ iid N(0,σ2 e) and {et} is independent of both Z1 and
Z2.
Prove that {Yt} is stationary.
Suppose that Z1 and Z2 are uncorrelated random variables with zero mean and unit variance. Consider...
Exercise 8.43. Let Z1, Z2,... . Zn be independent normal random variables with mean 0 and variance 1. Let (a) Using that Y is the sum of independent random variables, compute both the mean and variance of Y. (b) Find the moment generating function of Y and use it to compute the mean and variance of Y. Exercise 8.43. Let Z1, Z2,... . Zn be independent normal random variables with mean 0 and variance 1. Let (a) Using that Y...
5.30 Suppose two random variables Xand Yare both zero mean and unit variance. Furthermore, assume they have a correlation coefficient of p. Two new random variables are formed according to W = aX + bY, Z = cX + dY. Determine under what conditions on the constants a,b. c, and d the random variables W and Z are uncorrelated.
QUESTION4 (a) Let e be a zero-mean, unit-variance white noise process. Consider a process that begins at time t = 0 and is defined recursively as follows. Let Y0 = ceo and Y1-CgY0-ei. Then let Y,-φ1Yt-it wt-1-et for t > ï as in an AR(2) process. Show that the process mean, E(Y.), is zero. (b) Suppose that (a is generated according to }.-10 e,-tet-+扣-1 with e,-N(0.) 0 Find the mean and covariance functions for (Y). Is (Y) stationary? Justify your...
Suppose Z2-0Z1 +A where Z1, A are independent random variables with mean 0. atea rž-2, σ 1-1 and φ-0.1, what is σ ? Part b) If σ 1-6 and φ 0.3, what is ơå in order that σ σ , ?
2. Suppose that Ya ut where the ut are iid Normal with mean zero and variance σ2, but that you mistakenly think Yt is difference stationary. You therefore construct a new series a) Are the Xt i.i.d.? Explain b) Is X stationary? Explain c) Calculate the mean, variance, and autocorrelation function of X d) How does the answer you obtained in (c) compare with the mean, variance and autocor- relation function of Y? 2. Suppose that Ya ut where the...
onsider the process Y, = Y + Σ|e, where Yo ~ (μ, σ2) and the e's are 0-mean, a stationary process? independent identically distributed random variables with variance 1. Is (Y How about the process ▽Yǐ = Yt-)t-1 ? Explain. onsider the process Y, = Y + Σ|e, where Yo ~ (μ, σ2) and the e's are 0-mean, a stationary process? independent identically distributed random variables with variance 1. Is (Y How about the process ▽Yǐ = Yt-)t-1 ? Explain.
Let ˜x and ˜y be zero-mean, unit variance Gaussian random variables with correlation coefficients, . Suppose we form two new random variables using linear transformations: Let and be zero-mean, unit variance Gaussian random variables with correlation coefficients, p. Suppose we form two new random variables using linear transformations: Find constraints on the constants a, b, e, and d such that ù and o are inde- pendent.
Let X and Y be two independent Gaussian random variables with common variance σ2. The mean of X is m and Y is a zero-mean random variable. We define random variable V as V- VX2 +Y2. Show that: 0 <0 Where er cos "du is called the modified Bessel function of the first kind and zero order. The distribution of V is known as the Ricean distribution. Show that, in the special case of m 0, the Ricean distribution simplifies...
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. 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?