Stationary and Mean Ergodic for random processes

#1
I need to determine whether the random processes below are strict-sense stationary (SSS), whether it is wide-sense stationary (WSS), and whether it is ergodic in the mean. All help is much appreciated.

To my understanding, by definition, to be WSS, the mean and autocorrelation function need to be independent of time k. To be SSS, all statistics need to be independent of time, i.e. invariant to time delays. For mean-ergodic, the ensemble and time averages need to be equal. My issue is proving and justifying.

1666890472255.png
1666890491956.png
 
#2
My thoughts:

For part (a)
Since the Poisson RV includes a time duration in its PDF, then X_k would depend on time, so it is not WSS.
To test for SSS, I simply replace k with k - d , so X_(k-d) = e^-|k-d - D|, but since it is not WSS, then it would still dependent on time, so it is not SSS.
I'm not sure how to approach proving if the ensemble and time averages are the same or not. A time average is generally a RV and an ensemble average is generally a function of time. Any help appreciated.

As for (b)
The mean would be constant E(X_k) = (1/3)(1/3) = 1/9, but I'm not sure how to write out the autocorrelation function, but my hutch is that it's dependent of time, since it's equally likely to be 1, 2, or 3 -> random process is not WSS.
To test for SSS, a similar approach with a time delay as with part (a). I suspect the process is not SSS and depends on time k since {U_k} ~ i.i.d Unif(1,2,3)
I think that given the U_k, the ensemble and time average would both come out as 1/9, meaning that the process is mean-ergodic.