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If I have two random variables $X_n$ and $X_{n-1}$ where $X_n = \{{X_{n-1}, X_{n-1}-1\}} $ (can only have those two values). I also want to calculate the following expression:

$$P[X_{n+1} = i_n , X_{n} = i_n, X_{n-1} = i_{n-1},...] + P[X_{n+1} = i_n-1 , X_{n} = i_n, X_{n-1} = i_{n-1},...] \over P[X_{n} = i_n, X_{n-1} = i_{n-1},...] $$

Note that $X_n $ and $X_{n-1}$ have related value. Can I exploit this to make the above expression simpler?

cyberic
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  • Is there context to this problem? It seems like a Markov chains question - is there something specific you want to manipulate that expression to show? It seems to me like it simplifies to just $1$. – B. Mehta Apr 18 '18 at 02:09
  • @B.Mehta yes It's a Markov chain problem. I want to improve that this quantity above equals to $P[X_{n+1} =j |X_{n} = i]$ note that $j={ i, i-1}$ – cyberic Apr 18 '18 at 02:21

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