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\flushright{
Statistics \\
MATH 371\\
Homework 0\\
Part 1 of 2.}

\begin{exer}
\label{160912.1}
	Let $A_1,A_2,\dots$ be an arbitrary infinite sequence of events, and let $B_1,B_2,\dots$ be another infinite sequence of events defined as
	\begin{align*}
		B_1 &= A_1\\
		B_2 &= A_1^c \cap A_2\\
		B_3 &= A_1^c\cap A_2^c \cap A_2\\
		\vdots
	\end{align*}
	Prove that
	\[
		\operatorname{Pr}\left( \bigcup_{i=1}^{n} A_i \right) = \sum_{i=1}^{n} \operatorname{Pr}(B_i)
	\]
	for $n=1,2,\dots$, and that
	\[
		\operatorname{Pr}\left( \bigcup_{i=1}^{\infty} A_i \right) = \sum_{i=1}^{\infty}\operatorname{Pr}(B_i).
	\]
\end{exer}

\begin{exer}
	Let $A_1,A_2,\dots$ be an infinite sequence of events such that $A_1\subset A_2\subset \cdots$. Prove that
	\[
		\operatorname{Pr}\left( \bigcup_{i=1}^{\infty} A_i \right) = \lim_{n\to \infty} \operatorname{Pr}(A_n).
	\]
	\textbf{Hint:} Use Exercise \ref{160912.1}.
\end{exer}

\begin{exer}
	Let $A_1,A_2,\dots$ be an infinite sequence of events such that $A_1\supset A_2\supset \cdots$. Prove that
	\[
		\operatorname{Pr}\left( \bigcap_{i=1}^{\infty} A_i \right) = \lim_{n\to \infty} \operatorname{Pr}(A_n).
	\]
\end{exer}

\begin{exer}
	 A student takes a 5 question true-or-false quiz and, not having studied, randomly guesses the answer to each question. Suppose that a correct answer is worth 1 point and an incorrect answer is worth $-1$ point. Find the probability that the student
	 \begin{multicols}{2}
	 \begin{enumerate}[(a)]
	 \item gets 5 points.
	 \item gets $-5$ points.
	 \item gets 3 points.
	 \item gets 2 points.
	 \item gets a positive score.
	 \end{enumerate}
	 \end{multicols}
\end{exer}

\begin{exer}
	For any two events $A$ and $B$ with $\operatorname{Pr}(B)>0$, prove that $\operatorname{Pr}(A^c|B) = 1 - \operatorname{Pr}(A|B)$.
\end{exer}

\begin{exer}
	Suppose that $A$, $B$, and $C$ are three independent events such that $\operatorname{Pr}(A) = \frac{1}{4}$, $\operatorname{Pr}(B) = \frac{1}{3}$, and $\operatorname{Pr}(C) = \frac{1}{2}$.
	\begin{enumerate}[(a)]
	\item Determine the probability that none of these three events will occur.
	\item Determine the probability that exactly one of these events will occur.
	\end{enumerate}
\end{exer}

\begin{exer}
	Suppose that the pdf of a random variable $X$ is
	\[
		f(x) = 
		\begin{cases}
			cx^2 & \text{for } 1\leq x\leq 2,\\
			0 & \text{otherwise.}
		\end{cases}
	\]
	\begin{enumerate}[(a)]
	\item Find the value of the constant $c$ and sketch the pdf.
	\item Find the value of $\operatorname{Pr}(X>\frac{3}{2})$.
	\end{enumerate}
\end{exer}

\begin{exer}
	Let $f$ be the joint pf/pdf of random variables $X$ and $Y$, with $X$ discrete and $Y$ continuous. Prove that the marginal pf of $X$ is
	\[
		f_X (x) = \operatorname{Pr}(X  =x) = \int_{\bbr} f(x,y)\, dy \text{ for all } x,
	\]
	and the marginal pdf of $Y$ is
	\[
		f_Y (y) = \sum_{x} f(x,y), \text{ for } y\in \bbr.
	\]
\end{exer}

\end{document}


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