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\problem {\bf (5 points)} Let the random variable $R$ be equal to the
number of heads that come up when four mutually independent, fair
coins are flipped.

\begin{problemparts}
\problempart Compute the probability distribution function for $R$.
That is, determine $\Pr(R = k)$ for all $k$.

\solution{
\begin{eqnarray*}
\Pr(R = 0)	& = &	\frac{1}{16} \\
\Pr(R = 1)	& = &	\frac{4}{16} \\
\Pr(R = 2)	& = &	\frac{6}{16} \\
\Pr(R = 3)	& = &	\frac{4}{16} \\
\Pr(R = 4)	& = &	\frac{1}{16}
\end{eqnarray*}
}

\problempart Compute the cumulative distribution function for $R$.
That is, find $\Pr(R \leq k)$ for all $k$.

\solution{
\begin{eqnarray*}
\Pr(R \leq 0)	& = &	\frac{1}{16} \\
\Pr(R \leq 1)	& = &	\frac{5}{16} \\
\Pr(R \leq 2)	& = &	\frac{11}{16} \\
\Pr(R \leq 3)	& = &	\frac{15}{16} \\
\Pr(R \leq 4)	& = &	\frac{16}{16}
\end{eqnarray*}
}

\end{problemparts}