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There is an information source on the information source alphabet $A = \{a, b, c\}$ represented by the state transition diagram below:

Markov chain

a) The random variable representing the $i$-th output from this information source is represented by $X_i$. It is known that the user is now in state $S_1$. In this state, let $H (X_i|s_1)$ denote the entropy when observing the next symbol $X_i$, find the value of $H (X_i|s_1)$, entropy of this information source, Calculate $H (X_i|X_{i-1}) $ and $H (X_i)$ respectively. Assume $i$ is quite large

How can I find $H(X_i|s_1)?$ I know that $$H(X_i|s_1) = -\sum_{i,s_1} p\left(x_i, s_1\right)\cdot\log_b\!\left(p\left(x_i|s_1\right)\right) = -\sum_{i,j} p\left(x_i, s_1\right)\cdot\log_b\!\left(\frac{p\left(x_i, s_1\right)}{p\left(s_1\right)}\right)$$ but I don't know $p(s_1)$.

$$A=\begin{pmatrix}0.25 & 0.75 & 0\\0.5 & 0 & 0.5 \\0 & 0.7 & 0.3 \end{pmatrix}.$$

From matrix I can know that $p(s_1|s_1)=0.25$, etc.

But what is the probability of $s_1$? And how can I calculate $H (X_i|X_{i-1})$?

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  • $\begingroup$ Also posted on mathematics about an hour earlier. Please do not post the same question on multiple sites. Each community should have an honest shot at answering without wasting anybody's time. If you don't get a satisfying answer after a week or so, you may flag to request migration. $\endgroup$ – Apass.Jack Jul 16 at 19:55
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You can make the golloowing set of equations, where s1 - the probability of being in s1 state:

s1 = 0.25s1 + 0.5s2
...
s1+s2+s3=1

I don't know the theory, but think that entropy of each state can be computed as follows:

e1 = - 0.25*log(0.25) - 0.75*log(0.75)

with overall entropy given already computed probability and entropy of each state

e = e1*s1 + e3*s3 + e3*s3

I don't know what H(...) means.

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  • $\begingroup$ thankyou! is this right way to count entropy? $H(X_i|s_1) = -\sum_{i,s_1} p\left(x_i, s_1\right)\cdot\log_b\!\left(p\left(x_i|s_1\right)\right) = -\sum_{i,j} p\left(x_i, s_1\right)\cdot\log_b\!\left(\frac{p\left(x_i, s_1\right)}{p\left(s_1\right)}\right)$ ? and how can i find $H (X_i|X_{i-1})$? $\endgroup$ – devss Jul 19 at 15:52
  • $\begingroup$ @devss see my edit. $\endgroup$ – Bulat Jul 19 at 16:09

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