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{\large\bf Exercise sheet 03 -- 18.04.2024}
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\begin{bsp}
Let $X$ and $Y$ be the values of two independently thrown six-sided dice.
Calculate
\begin{enumerate}
    \item $H(X)+H(Y)$,
    \item $H(X+Y)$,
    \item $H(X-Y)$,
    \item $H(\max(X,Y))$.
\end{enumerate}
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\begin{bsp}
A fair coin is flipped until head occurs the first time. Let $X$ denote the number of flips required.
\begin{enumerate}
    \item Find the entropy $H(X)$ in bits.
    \item Find an efficient sequence of yes-no questions in order to guess the outcome of $X$. Compare $H(X)$ to the expected number of questions.
\end{enumerate}
\end{bsp}

\begin{bsp}
The play-offs of NBA are played between team A and team B in a Best-of-Seven series that terminates as soon as one of the teams wins four games. Let the random variable $X$ represent the outcome of the series of games (some possible values are AAAA, BABABAB and AAABBBB). Let $Y$ denote the number of games played. Assuming that the two teams are equally strong, determine the values of $H(X)$, $H(Y)$, $H(Y \mid X)$ and $H(X \mid Y)$.
\end{bsp}

\begin{bsp}
Let $X$ and $Y$ be random variables that take values in finite sets $\mathcal{X}$ and $\mathcal{Y}$, respectively. Let $Z = X + Y$.
\begin{enumerate}
    \item Show that $H(Z \mid X) = H(Y \mid X)$. Argue that if $X, Y$ are independent, then $H(Y) \leq H(Z)$ and $H(X) \leq H(Z)$.
    \item Give an example of (necessarily dependent) random variables in which $H(X) > H(Z)$ and $H(Y) > H(Z)$.
    \item Find necessary and sufficient conditions for $H(Z) = H(X) + H(Y)$.
\end{enumerate}
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