Let W1, W2, . . . be an uncorrelated random sequence with mean 0 and variance 1.Define the discrete-time random process {Xn : n ∈ N} := {X1, X2, . . .} by Xn = aXn−1 + Wn(n ∈ N) with a and X0 given. For each of the following two separate cases, find the mean functionmX (n) (n ∈ N) and covariance function CX (m, n) (m, n ∈ N) for the process {Xn : n ∈ N}, anddetermine if it is wide-sense stationary.a. a = 1 and X0 = 0.b. |a| < 1 and X0 is a random variable with mean 0 and variance 1/(1 − a2), uncorrelated withW1, W2, . . . .
Let W1, W2, . . . be an uncorrelated random sequence with mean 0 and variance 1.Define the discrete-time random process {Xn : n ∈ N} := {X1, X2, . . .} by Xn = aXn−1 + Wn(n ∈ N) with a and X0 given. For each of the following two separate cases, find the mean functionmX (n) (n ∈ N) and covariance function CX (m, n) (m, n ∈ N) for the process {Xn : n ∈ N}, anddetermine if it is wide-sense stationary.a. a = 1 and X0 = 0.b. |a| < 1 and X0 is a random variable with mean 0 and variance 1/(1 − a2), uncorrelated withW1, W2, . . . .
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
Problem 1CR
Related questions
Question
Let W1, W2, . . . be an uncorrelated random sequence with mean 0 and variance 1.
Define the discrete-time random process {Xn : n ∈ N} := {X1, X2, . . .} by Xn = aXn−1 + Wn
(n ∈ N) with a and X0 given. For each of the following two separate cases, find the meanfunction
mX (n) (n ∈ N) andcovariance function CX (m, n) (m, n ∈ N) for the process {Xn : n ∈ N}, and
determine if it is wide-sense stationary.
a. a = 1 and X0 = 0.
b. |a| < 1 and X0 is a random variable with mean 0 and variance 1/(1 − a2), uncorrelated with
W1, W2, . . . .
Define the discrete-time random process {Xn : n ∈ N} := {X1, X2, . . .} by Xn = aXn−1 + Wn
(n ∈ N) with a and X0 given. For each of the following two separate cases, find the mean
mX (n) (n ∈ N) and
determine if it is wide-sense stationary.
a. a = 1 and X0 = 0.
b. |a| < 1 and X0 is a random variable with mean 0 and variance 1/(1 − a2), uncorrelated with
W1, W2, . . . .
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