\problemdata     
{S01-PS10-1}           
{probability-intro}              
{} 		
{S98-PS10, S01-PS10}
{S01 Karger}

\begin{problem}



\bparts
\ppart%{\bf (5 points)}
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} :
\[
\begin{array}{|c|c|}
\hline
 \text{Value of }2 ^{R_1} R_2  & \text{Probability} \\
\hline
 \cdots &   \cdots    \\
 \cdots &   \cdots    \\
 \cdots &   \cdots    \\\hline
\end{array}
\]

\solution{


Let $X = 2 ^{R_1} R_2$. 

$\Pr(X = 1) = \Pr(R_1 = 0 \wedge R_2 = 1) = 1/4$.\\
$\Pr(X = 2) = \Pr((R_1 = 0 \wedge R_2 = 2)
  \vee (R_1 = 1 \wedge R_2 = 1) = 3/8$.\\
$\Pr(X = 4) = \Pr((R_1 = 0 \wedge R_2 = 4)\vee(R_1 \wedge R_2=2))=1/4$.\\
$\Pr(X = 8) = \Pr(R_1=1 \wedge R_2=4) = 1/8$.\\

To sum up:
\[
\begin{array}{|c|c|}
\hline
 \text{Value} & \text{Probability} \\
\hline
      1       &    \frac{1}{4}     \\
\hline
      2       &    \frac{3}{8}     \\
\hline
      3       &    \frac{1}{4}     \\
\hline
      4       &    \frac{1}{8}     \\
\hline
\end{array}
\]


}

\ppart%{\bf (5 points)}
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.

\solution{
{\bf Solution}

$\Pr(X = 1) = \Pr(R_2 = 1) = 1/2$.\\
$\Pr(X = 4) = \Pr(R_2 = 2) = 1/4$.\\
$\Pr(X = 8) = \Pr(R_2 = 4) = 1/4$.\\
So the final result is:
\[\begin{array}{|c|c|}
\hline
 \text{Value} & \text{Probability} \\
\hline
      1       &    \frac{1}{2}     \\
\hline
      4       &    \frac{1}{4}     \\
\hline
      8       &    \frac{1}{4}     \\
\hline
\end{array}
\]
}

\eparts


\end{problem}
