I am going through Rabiner 1989 and he writes that the discrete probability density function of duration $d$ in state $i$ (that is, staying in a state for duration $d$, conditioned on starting in that state) is $$p_i(d) = {a_{ii}}^{d-1}(1-a_{ii})$$

($a_{ii}$ is the state transition probability from state $i$ to state $i$ - that is, staying in the same state).

He then continues to say that the expected durations in a state, conditioned on starting in that state, is $$\overline d_i = \sum_{i=1}^\infty d p_i(d) = \frac{1}{1-a_{ii}}$$

Where does the coefficient $d$ (in $\sum_{i=1}^\infty d p_i(d)$) come from?

  • $\begingroup$ Looks like a typo and sum should be over $d$, not $i$, in the second equation. But I am not familiar with that tutorial. $\endgroup$ – Neil Slater Mar 2 '20 at 7:44
  • $\begingroup$ @NeilSlater Still, even if it is summing over $d$, what does the coefficient $d$ represent? Is it adding $p_i(d)$ each time it stays in state $i$? $\endgroup$ – jstycrpsc Mar 2 '20 at 20:08
  • $\begingroup$ You already state that: duration 𝑑 $\endgroup$ – Neil Slater Mar 2 '20 at 22:04
  • $\begingroup$ @NeilSlater Sorry I was not clear. Why do we multiply $p_i(d)$ by $d$? $\endgroup$ – jstycrpsc Mar 3 '20 at 4:54
  • $\begingroup$ . . . .because it is taking a mean/expected value by multiplying each possible value by its probability of occurring $\endgroup$ – Neil Slater Mar 3 '20 at 6:47

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