Random sequence question

[rsingh628]

Hello all, I have a random sequences question and I am mostly struggling with the last part (e) with deriving the marginal pdf. Any help would be greatly appreciated.
My attempt for the other parts a - d is also below, and it would nice if I can get the answers confirmed to ensure I'm understanding things properly.


Definition of ergodic being used:


Attempt:
For part (a) I believe it is yes, because the mean is 0 (constant) and the autocorrelation function is independent of time k. I got Rx(m) = 0.5*cos(0.2*pi*m)
For (b) I think it is yes because all statistics are not dependent on time k.
For (c) both the ensemble and time averages would be 0, and since these are equal it seams yes, Xk is ergodic in the mean.
For (d), I believe it is no, because the autocorrelation function is a periodic sinusoid and goes on infinitely, so the limit as Rx(m) goes to infinity does not exist, i.e. it is not constant and not equal to u^2

 

[blamocur]

Can you figure out CDF for [imath]X_k[/imath] ? I.e., [imath]P(\cos (0.2\pi k+\theta) \leq x[/imath] ?

 

[rsingh628]

Can you figure out CDF for [imath]X_k[/imath] ? I.e., [imath]cos (0.2\pi k+\theta) \leq x[/imath] ?

Yes, could start with the CDR and differentiate to find the PDF, but I'm not sure how to manipulate to fit the definition of the CDF [imath]P(X_k \leq x )[/imath]
Next step would be:
[imath]P( (0.2\pi k+\theta) \leq arccos(x) )[/imath]

 

[blamocur]

Yes, could start with the CDR and differentiate to find the PDF, but I'm not sure how to manipulate to fit the definition of the CDF [imath]P(X_k \leq x )[/imath]
Next step would be:
[imath]P( (0.2\pi k+\theta) \leq arccos(x) )[/imath]

Almost: you have to remember that [imath]\cos[/imath] is not a monotonic function.