\psol{11}{December 11, 1998}

\newcommand{\Ex}{\text{Ex}}

% \centerline{\bf Read sections 4.4 and 4.5 in Rosen.}

\begin{problems}

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\problem {\bf (10 points)} 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}
\problempart $\Pr(A \cup B) = \Pr(A) + \Pr(B)$

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

\problempart $\Pr(A \cup B) = \Pr(A) + \Pr(B) - \Pr(A \cap B)$

\solution{
This statement is always true.
}

\problempart $\Pr(A \cap B) = \Pr(A) \cdot \Pr(B)$

\solution{
This statement is true provided that events $A$ and $B$ are independent.
}

\problempart $\Pr(A \cap B) < \Pr(A)$

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

\problempart $\Ex(S + T) = \Ex(S) + \Ex(T)$

\solution{
This statement is always true.
}

\problempart $\Ex(S \cdot T) = \Ex(S) \cdot \Ex(T)$

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

\end{problemparts}