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I have a question that asks me "What is the expected number of observations in a state?" with the note: $$\sum^{\infty}_{d=1}d a^{d=1} = \frac{1}{(a-1)^2}\text{ when } |a| < 1$$ Prior to that I have a Markov Model with 3 states and the transition probabilities as follows:

  • S1: $P(S1) = 0.4$, $P(S2|S1) = 0.3$, $P(S3|S1) = 0.3$
  • S2: $P(S2) = 0.6$, $P(S1|S2) = 0.2$, $P(S3|S2) = 0.2$
  • S3: $P(S3) = 0.8$, $P(S1|S3) = 0.1$, $P(S2|S3) = 0.1$

Any guidance on what the note means or what I am trying to work out would be appreciated.

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The note probably refers to the calculation of the expected value of a random variable $T_{i}$ that describes the time "we stay" in the state $S_i$.

From how I understand the question, this is exactly what you are tasked to compute. Therefore I understood the question as: "given that you are in some state $S_i$, how many observations of $S_i$ do you expect before changing a state?"

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  • $\begingroup$ So I imagine for $S1$ you would expect $\approx 3$, $S2 \approx 6$ and $S3 \approx 25$? $\endgroup$ Feb 27, 2022 at 15:04
  • $\begingroup$ I think you are correct, assuming this really is the intention of the question $\endgroup$
    – nir shahar
    Mar 2, 2022 at 18:59

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