
\problemdata     
{S01-PS10-2}           
{probability: various}              
{} 		
{F00-PS11, S01-PS10}
{S01 Karger}


\begin{problem}


 Several statements involving probability
are listed below.  Assume that $A$ and $B$ are events, $S$ and $T$ are
random variables, and the underlying sample space is countable.
Furthermore, assume that all expectations exist.  For each statement,
describe restrictions on $A$, $B$, $S$, and $T$ sufficient to ensure
that the statement is true.  Make your restrictions as minimal as you
can.  No proofs are necessary.

\begin{problemparts}
%\begin{enumerate}

\problempart $\Pr(A \cup B) = \Pr(A) + \Pr(B)$
%\item[\textbf{(a)}] $\Pr(A \cup B) = \Pr(A) + \Pr(B)$

\solution{
\textbf{Solution:}\\
This statement is true provided that the event $A \cap B$
has probability zero.}

\problempart $\Pr(A \cup B) = \Pr(A) + \Pr(B) - \Pr(A \cap B)$
%\item[\textbf{(b)}] $\Pr(A \cup B) = \Pr(A) + \Pr(B) - \Pr(A \cap B)$

\solution{
\textbf{Solution:}\\
This statement is always true.
}

\problempart $\Pr(A \cap B) = \Pr(A) \cdot \Pr(B)$
%\item[\textbf{(c)}] $\Pr(A \cap B) = \Pr(A) \cdot \Pr(B)$

\solution{
\textbf{Solution:}\\
This statement is true provided that events $A$ and $B$ are independent.
}
\problempart $\Pr(A \cap B) \leq \Pr(A)$
%\item[\textbf{(d)}] $\Pr(A \cap B) < \Pr(A)$

\solution{
\textbf{Solution:}\\
This statement is always true.
}
\problempart $\Pr(A \cap B) < \Pr(A)$
%\item[\textbf{(d)}] $\Pr(A \cap B) < \Pr(A)$

\solution{
\textbf{Solution:}\\
This statement is true provided $\Pr(A \cap \overline{B}) > 0$.
}

\problempart $\Ex(S + T) = \Ex(S) + \Ex(T)$
%\item[\textbf{(e)}] $\Ex(S + T) = \Ex(S) + \Ex(T)$

\solution{
\textbf{Solution:}\\
This statement is always true.
}

\problempart $\Ex(S \cdot T) = \Ex(S) \cdot \Ex(T)$
%\item[\textbf{(f)}] $\Ex(S \cdot T) = \Ex(S) \cdot \Ex(T)$

\solution{
\textbf{Solution:}\\
This statement is true provided that random variables $S$
and $T$ are independent.}

\end{problemparts}
\end{problem}
