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I am considering the discrete-time discrete-valued random process $X_n$ that consists of the sequences

Sequence1:   0 0 1 0 0 1 0 0 1 0 0 1 0 0 ...
Sequence2:   1 0 0 1 0 0 1 0 0 1 0 0 1 0 ...
Sequence3:   0 1 0 0 1 0 0 1 0 0 1 0 0 1 ...

with each sequence being chosen with equal probability $\frac{1}{3}$.

After determining that this process is not an i.i.d. random process, I am having trouble finding the joint PMF of $X_{n_1}$ and $X_{n_2}$. Any help would be greatly appreciated. Thanks!

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Hint: The random variables are identically distributed as Bernoulli random variables with parameter $\frac{1}{3}$, but wouldn't you say that $X_n$ and $X_{n+3}$ are equal (with probability $1$ if you like) and so are not independent? –  Dilip Sarwate Dec 13 '12 at 23:32
    
@DilipSarwate I see, what then would be a way to find the PMF of $X_n$? –  αδζδζαδζασεβ23τ254634ω5234ησςε Dec 14 '12 at 1:39
    
>what then would be a way to find the PMF of $X_n$? The PMF of $X_n$ was specified completely in the first sentence of my previous comment, but to cross the $i$'s and dot the $t$'s $$p_{X_n}(0) = \frac{2}{3}, ~~ p_{X_n}(1) = \frac{1}{3} ~~ \text{for all} ~ n$$ can be obtained by inspection. –  Dilip Sarwate Dec 14 '12 at 3:23
    
@DilipSarwate Oh okay, I believe I am now beginning to understand the idea of describing the PMF of a random process. But for the joint PMF, we would have to consider the certain repetitiveness of the sequence then? –  αδζδζαδζασεβ23τ254634ω5234ησςε Dec 14 '12 at 4:12
    
See also math.stackexchange.com/questions/258588. –  joriki Dec 14 '12 at 9:56
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