Q1
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Probability
└── Homework
└── W9
└── Q1.tex
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\fancyhead[l]{Li Yifeng}
\fancyhead[c]{Homework \#9}
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\begin{document}
\section*{Question 1}
\noindent Calculate the entropy in bits for each of the following random variables:
\bigskip
\begin{enumerate}[start=1,label={\bfseries Part \arabic*:},leftmargin=0in]
\bigskip\item Pixel values in an image whose possible grey values are all the integers from $0$ to $255$ with uniform probability.
\subsection*{Solution}
\[256 = 255 - 0 + 1\]
\[H(X) = \bE(I(X)) = -256 \cdot \frac{1}{256}\left(log_2\left(\frac{1}{256}\right)\right) = 8 \text{bits}\]
\subsection*{Answer}
\[\boxed{
H(X) = 8 \text{bits}
}\]
\bigskip\item Gender in a tri-sexed insect population whose three genders occur with probabilities $\frac{1}{4}$, $\frac{1}{4}$, and $\frac{1}{2}$.
\subsection*{Solution}
\[H(X) = \bE(I(X)) = - \left(
2 \cdot \frac{1}{4}\left(log_2\left(\frac{1}{4}\right)\right)
+ \frac{1}{2}\left(log_2\left(\frac{1}{2}\right)\right)
\right) = 1.5 \text{bits}\]
\subsection*{Answer}
\[\boxed{
H(X) = 1.5 \text{bits}
}\]
\bigskip\item A population of persons classified by whether they are older, or not older, than the population's median age.
\subsection*{Solution}
\[H(X) = \bE(I(X)) = - \left(
2 \cdot \frac{1}{2}\left(log_2\left(\frac{1}{2}\right)\right)\right) = 1 \text{bits}\]
\subsection*{Answer}
\[\boxed{H(X) = 1 \text{bits}}\]
\end{enumerate}
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