The relation between the input X (t) and Y (t) of a system is Y (t) = X4 (t). X (t) is a zero mean stationary Gaussian random process with auto correlation function Ryx (T) = e- a ld for a > 0. Find E[Y (t)] and Ryy (T). %3D
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![The relation between the input X (t) and Y (t) of a system is Y (t) = X4 (t). X (t) is
a zero mean stationary Gaussian random process with auto correlation function
RYx (T) = eaa for a > 0. Find E[Y (t)] and Ryy (t).
%3D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F281ed553-1571-4c44-bfa9-8d34fa1661ad%2Fdaf82797-597a-4a3a-9c81-ba8aa834945c%2Fggefbf5_processed.jpeg&w=3840&q=75)
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- Consider the geometric Brownian motion with σ = 1: dS = μSdt + SdX, and consider the function F(S) = A + BSα. Find any necessary conditions on A, B, and α such that the function F(S) follows a stochastic process with no drift.Which of the following processes (Xt)t is weakly stationary? A: Xt = 1:6 + Xt 1 + V tB: Xt = 0:6 Xt-1 +V tC: Xt = 0:8 Xt-1 + V tD: Xt = 0:8 t + 0:6 V t – 1 The term (t) is always assumed to be white noise with variance oneLet yt = φyt−1 + et with et ∼ WN(0,σ2) and |φ| < 1. Consider the over-differenced process wt = (1 − L)yt.(i) What is the model followed by wt? (ii) Is wt invertible? (iii) Obtain V [wt] and compare its magnitude with V [yt] and hence comment on the impact of over-differencing on the variance of a stationary process.
- LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.The joint pmf of X and Y is f(x, y) = 1/6, 0 ≤ x+y ≤ 2, where x and y are nonnegative integers. Compute Cov(X, Y) and determine the correlation coefficient.f X1,X2,...,Xn constitute a random sample of size n from a geometric population, show that Y = X1 + X2 + ···+ Xn is a sufficient estimator of the parameter θ.
- If we let RX(t) = ln MX(t), show that R X(0) = μ and RX(0) = σ2. Also, use these results to find the mean and the variance of a random variable X having the moment-generating function MX(t) = e4(et−1)Consider a random process X(t) defined by X(t) = U cos t + (V + 1) sin t, −∞ < t < ∞where U and V are independent random variables for which E(U) = E(V) = 0 E(U2) = E(V2) = 1(a) Find the autocovariance function KX(t, s) of X(t).(b) Is X(t) WSS?Find the maximum likelihood estimator for θ in the pdf f(y; θ) = 2y/(1 − θ^2), θ ≤ y ≤ 1.
- Use the expected value properties to obtain the E[Y] of the following system : y = 3x + 1 , where E[X] =3Let X and Y have the following joint distribution:X \Y 0 1 0 0.40 0.10 1 0.10 0.10 2 0.10 0.20 (a) Find Cov(4 + 2X , 3 − 2Y ). (b) Let Z = 3X − 2Y + 2. Find E[Z] and σ2z (c) Calculate the correlation coefficient between X and Y. What does this suggest about the relationship between X and Y? (d) Show that for two nonzero constants a and b, Cov(X + a, Y + b) = Cov(X , Y ).LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise (a) Show that the moment generating function mX(s) :=E(esX) =λ/(λ−s) for s< λ;