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From Ross Ch. 4 Ex. 45 (c):

Consider an irreducible finite Markov chain with states $0,1,\dots,N$.

Let $x_i=P\{$visit state $N$ before state $0$ | start in $i\}$. Then

$$x_i=\sum_{j=0}^{N}P_{ij}x_j,\qquad x_0=0,\qquad x_N=1$$

If $\sum_jjP_{ij}=i$ for $i=1,\dots,N-1$, show that $x_i=i/N$ is a solution to the above equations.

What does $\sum_jjP_{ij}=i$ even mean? I'm having trouble making any sense out of it. How can the summation of something be a state?

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hokoxixe
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  • It's not a state, it's an integer. (It's true that the states are numbered state 1, state 2, etc. but that's not what the equation means.) – saulspatz Mar 07 '19 at 14:23
  • Think of the numbers as numbers and not states. So $i$ is not a state,it is a number. The equations then make sense. – Sarvesh Ravichandran Iyer Mar 07 '19 at 14:27
  • It means that the 'expected' state you arrive in is state $i$ where in the expectation it is necessary to view the states as integers from $0$ to $N$. – Stan Tendijck Mar 07 '19 at 14:29

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