<p>I am going through <a href="https://ieeexplore.ieee.org/document/18626" rel="nofollow noreferrer">Rabiner 1989</a> and he writes that the discrete probability density function of duration <span class="math-container">$d$</span> in state <span class="math-container">$i$</span> (that is, staying in a state for duration <span class="math-container">$d$</span>, conditioned on starting in that state) is <span class="math-container">$$p_i(d) = {a_{ii}}^{d-1}(1-a_{ii})$$</span></p>

<p>(<span class="math-container">$a_{ii}$</span> is the state transition probability from state <span class="math-container">$i$</span> to state <span class="math-container">$i$</span> - that is, staying in the same state).</p>

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

<p>Where does the coefficient <span class="math-container">$d$</span> (in <span class="math-container">$\sum_{i=1}^\infty d p_i(d)$</span>) come from?</p>

CC BY-SA 4.0
2020-03-02T07:44:40.723Z
Looks like a typo and sum should be over $d$, not $i$, in the second equation. But I am not familiar with that tutorial.
Neil Slater
CC BY-SA 4.0
2020-03-02T20:08:54.163Z
@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 $i$?
jstycrpsc
CC BY-SA 4.0
2020-03-02T22:04:23.243Z
You already state that: duration 
Neil Slater
CC BY-SA 4.0
2020-03-03T04:54:54.273Z
@NeilSlater Sorry I was not clear. Why do we multiply $p_i(d)$ by $d$?
jstycrpsc
CC BY-SA 4.0
2020-03-03T06:47:29.463Z
. . . .because it is taking a mean/expected value by multiplying each possible value by its probability of occurring
Neil Slater
<math><probability><hidden-markov-model>
Expected duration in a state
jstycrpsc
# Expected duration in a state

(Asked by: username_0 on 2020-03-02)

I am going through [Rabiner 1989](https://ieeexplore.ieee.org/document/18626) 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?

**Tags:** math, probability, hidden-markov-model

> **Comments:**
> **username_1**: Looks like a typo and sum should be over $d$, not $i$, in the second equation. But I am not familiar with that tutorial.  
> **username_0**: @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 $i$?  
> **username_1**: You already state that: duration   
> **username_0**: @NeilSlater Sorry I was not clear. Why do we multiply $p_i(d)$ by $d$?  
> **username_1**: . . . .because it is taking a mean/expected value by multiplying each possible value by its probability of occurring  

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## 0 answers

