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A feed that drops ticks

StatisticsEasy

A tick $X$ is uniform on $16$ values. The feed $Y$ delivers it exactly, except that $15%$ of ticks are dropped, independently of $X$ , and the receiver can see a drop. What is $I (X; Y)$ in bits? Give a fraction or an exact decimal.

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Hint

Delivered ticks reveal all of $X$ , dropped ones nothing.

Solution

A delivered tick reveals $log_{2} 16 = 4$ bits.
A share of $1 - 15% = 0.85$ of ticks arrive, so $I (X; Y) = 0.85 \times 4 = 3.4 = \frac{17}{5}$ .

Answer: $\frac{17}{5}$

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