Recall that at time t the Poisson process N(t) is a Poisson random variable with parameter Xt. We can think of N(t) as counting the number of points in an interval (0, t] if we start at 0 as the origin (the meaning of these points depends on the application). Now define a new random process as X(t) = 1 if N(t) is even and X(t) = − 1 if N(t) is odd. Note that N(0) = 0 so X(0) = 1. i. Compute E[X(t)]. Simplify your result as much as possible. ii. As noted above X(0) = 1 (since N(0) = 0) and is not random. To remove this certainty we form a new random process Y(t) = A · X(t) where A is a random variable taking values +1 and −1 with equal probability. One can show that Y(t) is WSS with covariance function Ky(7) = e−²\|7|. Determine (mathematically) whether or not Y(t) is ergodic in mean.
Recall that at time t the Poisson process N(t) is a Poisson random variable with parameter Xt. We can think of N(t) as counting the number of points in an interval (0, t] if we start at 0 as the origin (the meaning of these points depends on the application). Now define a new random process as X(t) = 1 if N(t) is even and X(t) = − 1 if N(t) is odd. Note that N(0) = 0 so X(0) = 1. i. Compute E[X(t)]. Simplify your result as much as possible. ii. As noted above X(0) = 1 (since N(0) = 0) and is not random. To remove this certainty we form a new random process Y(t) = A · X(t) where A is a random variable taking values +1 and −1 with equal probability. One can show that Y(t) is WSS with covariance function Ky(7) = e−²\|7|. Determine (mathematically) whether or not Y(t) is ergodic in mean.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![b. Recall that at time t the Poisson process N(t) is a Poisson random
variable with parameter At. We can think of N(t) as counting the
number of points in an interval (0, t] if we start at 0 as the origin (the
meaning of these points depends on the application). Now define a new
random process as X(t) = 1 if N(t) is even and X(t) = -1 if N(t) is
odd. Note that N(0) = 0 so X (0) = 1.
www.
i. Compute E[X(t)]. Simplify your result as much as possible.
ii. As noted above X(0) = 1 (since N(0) = 0) and is not random.
To remove this certainty we form a new random process Y(t) =
A. X(t) where A is a random variable taking values +1 and -1
with equal probability. One can show that Y(t) is WSS with
covariance function Ky(T) = e-27. Determine (mathematically)
whether or not Y(t) is ergodic in mean.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03c3956a-8003-44d2-9b2a-84d1edb1c090%2F87a9c194-1166-4c29-8e33-91b2a236379c%2Ft998uo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:b. Recall that at time t the Poisson process N(t) is a Poisson random
variable with parameter At. We can think of N(t) as counting the
number of points in an interval (0, t] if we start at 0 as the origin (the
meaning of these points depends on the application). Now define a new
random process as X(t) = 1 if N(t) is even and X(t) = -1 if N(t) is
odd. Note that N(0) = 0 so X (0) = 1.
www.
i. Compute E[X(t)]. Simplify your result as much as possible.
ii. As noted above X(0) = 1 (since N(0) = 0) and is not random.
To remove this certainty we form a new random process Y(t) =
A. X(t) where A is a random variable taking values +1 and -1
with equal probability. One can show that Y(t) is WSS with
covariance function Ky(T) = e-27. Determine (mathematically)
whether or not Y(t) is ergodic in mean.
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