A coin is biased , such that the probability of getting Head is 2/3 and that of Tail is 1/ 3. This coin is tossed repeatedly until a Head appear for the first time. let L be the number of tosses to get this first Head. The entropy H(L) in bits is ________
1.38

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Solution

The correct option is A 1.38
The probability associate with each value of L in the given experiement are tabulated as shown below :

L 1 2 3 4 ...

Probability $\frac{2}{3}$ $(\frac{1}{3}) (\frac{2}{3}) = \frac{2}{3^{2}}$ ${(\frac{1}{3})}^{2} (\frac{2}{3}) = \frac{2}{3^{3}}$ ${(\frac{1}{3})}^{3} (\frac{2}{3}) = \frac{2}{3^{4}}$ ...

$H (L) = E n t r o p y {\frac{2}{3}, \frac{2}{3^{2}}, \frac{2}{3^{3}}, \frac{3}{3^{4}}, . . . .}$
$= \infty \sum t = 1 (\frac{2}{3^{t}}) l o g_{2} (\frac{2}{3^{i}}) b i t s$
$= \infty \sum t = 1 (\frac{2}{3^{t}}) l o g_{2} (2) + \infty \sum i = 1 (\frac{2}{3^{i}}) l o g_{2} (3^{i}) b i t s$

$= 2 l o g_{2} (3) [\infty \sum i = 1 (\frac{i}{3^{i}})] - 2 l o g_{2} (2) [\infty \sum i = 1 (\frac{1}{3^{i}})] b i t s$

$= \frac{2}{3} l o g_{2} (3) [\infty \sum i = 1 (\frac{i}{3^{i} - 1})] - \frac{2}{3} l o g_{2} (2) [\infty \sum i = 1 (\frac{1}{3^{i}})] b i t s$

$= \frac{2}{3} l o g_{2} (3) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{1}{{(1 - \frac{1}{3})}^{2}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ - \frac{2}{3} l o g_{2} (2) ⎡ ⎢ ⎢ ⎢ ⎣ \frac{1}{1 - \frac{1}{3}} ⎤ ⎥ ⎥ ⎥ ⎦ b i t s$

$= (\frac{2}{3} \times \frac{9}{4}) l o g_{2} (3) - (\frac{2}{3} \times \frac{3}{2}) l o g_{2} (2) b i t s$
$= [\frac{3}{2} l o g_{2} (3) - 1] b i t s \approx 1.38 b i t s$

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L	1	2	3	4	...
Probability	$\frac{2}{3}$	$(\frac{1}{3}) (\frac{2}{3}) = \frac{2}{3^{2}}$	${(\frac{1}{3})}^{2} (\frac{2}{3}) = \frac{2}{3^{3}}$	${(\frac{1}{3})}^{3} (\frac{2}{3}) = \frac{2}{3^{4}}$	...

A coin is biased , such that the probability of getting Head is 2/3 and that of Tail is 1/ 3. This coin is tossed repeatedly until a Head appear for the first time. let L be the number of tosses to get this first Head. The entropy H(L) in bits is ________ 1.38

A coin is biased , such that the probability of getting Head is 2/3 and that of Tail is 1/ 3. This coin is tossed repeatedly until a Head appear for the first time. let L be the number of tosses to get this first Head. The entropy H(L) in bits is ________
1.38