Information theory given by tables:XiPiThe probabilities of states of independent systems X and Y areX1=1X2=-2Y2=00.30.70.4System Z is related to systems X and Y by the relation Z = X + (X-Y)*2.Find complete information I(Z, Y) about system Y contained in system Z.YjRjY1=-10.6

Answered: given by tables: Xi Pi The…

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ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

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Information theory

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Step 1

Given the probabilities of states of independent systems X and Y as

$X_{i}$	$X_{1} = 1$	$X_{2} = - 2$
$P_{i}$	0.3	0.7

$Y_{j}$	$Y_{1} = - 1$	$Y_{2} = 0$
$R_{j}$	0.6	0.4

System Z is related to systems X and Y by the relation, $Z = X + {(X - Y)}^{2}$

If $X_{1} = 1 a n d Y_{1} = - 1 \Rightarrow Z = 5$

$X_{1} = 1 a n d Y_{2} = 0 \Rightarrow Z = 2$

$X_{2} = - 2 a n d Y_{1} = - 1 \Rightarrow Z = - 1$

$X_{2} = - 2 a n d Y_{2} = 0 \Rightarrow Z = 2$

The probabilities of the states of the system Z :

$z_{k}$	$z_{1} = - 1$	$z_{2} = 2$	$z_{3} = 5$
$q_{k}$	$0.7 \times 0.6 = 0.42$	$(0.3 \times 0.4) + (0.7 \times 0.4) = 0.4$	$0.3 \times 0.6 = 0.18$

The probabilities of the states of the combined system $(Y, Z)$

$η (p) = p \times \log_{2} (1 / p)$

$H (Y) = η (0.6) + η (0.4) = 0.44388 + 0.52876 = 0.97264$

$H (Z) = η (0.42) + η (0.4) + η (0.18) = 0.52542 + 0.52876 + 0.445518 = 1.4997$

$H (Y, Z) = η (0.42) + η (0.18) + η (0.4) = 1.4997$

$\begin{array}{rcl} I (Z, Y) & = & H (Z) + H (Y) - H (Y, Z) \\ = & 1.4997 + 0.97264 - 1.4997 \\ = & 0.97264 \end{array}$

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