Q3
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Probability
└── Homework
└── W9
└── Q3.tex
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\begin{document}
\section*{Question 3}
\noindent Suppose that $X$ is a random variable whose entropy $H(X)$ is $8$ bits. Suppose that $Y(X)$ is a deterministic function that takes on a different value for each value of $X$.
\bigskip
\begin{enumerate}[start=1,label={\bfseries Part \arabic*:},leftmargin=0in]
\bigskip\item What is the entropy of $Y$?
\subsection*{Solution}
Since the values are unrelated to the info or the entropy, easy to see that
\[H(X) = 8\text{bits} = H(Y)\]
\subsection*{Answer}
\[\boxed{H(Y) = 8 \text{bits}}\]
\bigskip\item What is the joint entropy $H(X,Y)$?
\subsection*{Solution}
Similarly, since both entropies are the same, we have
\[H(X, Y) = H(X) = H(Y) = 8 \text{bits}\]
\subsection*{Answer}
\[\boxed{H(X, Y) = 8 \text{bits}}\]
\end{enumerate}
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