10. Is it possible for X, Y, and Z to have the same distribution and satisfy X = U(Y + Z), whereU is uniform on [0, 1], and Y, Z are independent of U and of one another? (This question arises inmodelling energy redistribution among physical particles.)

Answered: Image /qna-images/answer/150c1fbf-57d9-44aa-a7b5-fb2ac19dd30c.jpg

Answered: 10. Is it possible for X, Y, and Z to…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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b) Let X(t) be a zero mean, stationary, Gaussian process with autocorrelation function R (T). This process is applied to a square law device defined by the input-output relation Y(t) = X(t), where Y(t) is the output. i) Show that the mean of Y(1) is R (0). ii) Find out the autocovariance function of y(t).
7. For what points (xo,Yo) does Theorem A imply that the initial value problem y' =y\y\, _y(x)=y, has a unique solution on some interval |x – xo|
ith leverage value (hat-value) = (H)ii= hii, ith diagnal entry of hat matrix • The h¡¡ is considered to be large if hii > 2 * h = 2 * k + 1 = # of parameters, k + 1 n n = # of sample, where Ehii = k+1 h = average of diagonal entry.
2. Let X = 8 + Xt-1 + Wt-1, for t=1,2,...., with Xo=0, where {W}~WN(0,0²). Note when the drift term (8) is zero, X is a random walk. We can also write Xt = St + Σ=1 W₁, t = 1,2 ... a) Derive ux (t) and yx (t + h, t) b) Is {X} stationary? Why/Why not?
Answer the next two questions based on the information given in the following Exhibit: Exhibit: The stock price S of an underlying security follows the geometric Brownian motion process ds = μS dt+oS dz, where z is a standard Wiener process. Then, 4. the process followed by S4 is a. dS4 = (4μ+0²)S4dt +40S³ dz ds4 = (4μ+602)S4 dt + 40S4 dz c. ds4 = us dt + os dz b. d. ds4 = (+40)S^dt +oSdz 5. the process followed by G(S, 1) = e(T-t)/S² is a. dG = (30²-2μ - r)Gdt - 20Gdz b. dG = (2μ-r)dt - 20Gdz c. dG = (30²-2μ r)Gdt + Gdz 1 16
A Random process X(t) is applied to a network with impulse response h(t) = u(t) exp (-bt) where b > 0 is a constant. The cross- correlation of X(t) with the output Y(t) is known to have the same form, Ryy(t) = u(t)t exp(-bt) (i) Find the auto-correlation of Y(t).
Q.3 Let X(t), t≥ 0; denotes the Brownian motion process with drift and variance parameters, μ and 2, respectively. Then prove that Var(X(t) - X(0)) = to² as the time unit A → 0.
2. Let P denote the path that goes in straight lines from (1,1,1) to (-1,-1,1) to (-1,-1,-1) to (1,1,-1) and then back to (1,1, 1). 2xyz+z el +2 x²z+sin(z) Compute fp F. dr, where F(x, y, z) = P
Suppose that Y, follows the AR(p) model Y₁ =B₁ + B₁Y ₁-1 +...+BpY₂-p+ut, where E(utlYt-1, Yt-2, ...) = 0. Let Yt + hit = E(Yt+nlYt₁ Y₁-1). Yt + hlt can be calculated as: OA. Yt + hlt=Bo + B₁Yt+h-11t+...+BpYt+h-plt for h> p. OB. Yt+ht=(Bo+B₁ + ... + Bp) × (Y₂+ Y₁+h-11t+...+Yt+h-plt). OC. Yt + hlt=B₁ + B₁ + ... + Bp) Yt + h-11t OD. Yt + hlt Bo + B₁₂ Y ₁-₁ + ... + B₂Yt-p²
2. Consider a stationary AR(2) process y: = 0.6y-1 + 0.3y-2+ 8, where &~ WN(0, 4). a. Find the mean and variance of y:t. b. What would the ACF and PACF for this process look like? (You do not need to calculate the ACF and PACF, simply consider what shape is might have given the class of model from which it is drawn.)
Tr.28.
2. Use Wronskian to show that fi = x, f2 = ln x, and f3 = x ln x are C (R). linearly independent in

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10. Is it possible for X, Y, and Z to have the same distribution and satisfy X = U(Y + Z), where U is uniform on [0, 1], and Y, Z are independent of U and of one another? (This question arises in modelling energy redistribution among physical particles.)