\problemdata
{random variable}
{s01 ps10}

\begin{problem}
\begin{problemparts}
\problempart
Let $R_1$ and $R_2$ be two independent random variables.
$R_1$ takes the values $0,1$ with equal probability.
$R_2$ has the following distribution.
\[
\begin{array}{|c|c|}
\hline
 \text{Value of }R_2 & \text{Probability} \\
\hline
           1           &    \frac{1}{2}     \\
\hline
           2           &    \frac{1}{4}     \\
\hline
           4           &    \frac{1}{4}     \\
\hline
\end{array}
\]


Compute the distribution of $2 ^{R_1} R_2$. Give your answers in a table, as shown below\footnote{You should not assume that your answer must necessarily have 3 rows} :
\[
\begin{array}{|c|c|}
\hline
 \text{Value of }2 ^{R_1} R_2  & \text{Probability} \\
\hline
 \cdots &   \cdots    \\
 \cdots &   \cdots    \\
 \cdots &   \cdots    \\\hline
\end{array}
\]

\problempart
Now suppose that $R_1$ and $R_2$ are no longer independent.
In particular, $R_1 = 0$ if $R_2$ is 1, and $R_1 =1$ when $R_2$ is 2 or 4.
$R_2$ has the same distribution as in part (a).

Compute the distribution of $2^{R_1} R_2$. Give your answers in a table, as you did above.

\end{problemparts}
\end{problem}