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\textbf{\Large{Mathematical Statistics (MATH-UA 234)}}
\textbf{\Large{Homework 1}}
Due 09/15 at the beginning of class on \href{https://www.gradescope.com/courses/414277/}{Gradescope}
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\extrafootertext{problems with a textbook reference are based on, but not identical to, the given reference}
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\begin{problem}[Wasserman 1.19 (Bayes theorem)]
Suppose that 30 percent of computer owners use a Macintosh, 50 percent
use Windows, and 20 percent use Linux. Suppose that 65 percent of
the Mac users have succumbed to a computer virus, 82 percent of the
Windows users get the virus, and 50 percent of the Linux users get
the virus. We select a person at random and learn that her system was
infected with the virus. What is the probability that they are a Windows
user?
\end{problem}
\begin{problem}
Suppose $\PP[A] = 2/3$ and $\PP[B^c] = 1/4$. Can $A$ and $B$ be disjoint (only one can occur)? Why or why not?
\end{problem}
\begin{problem}
Use the axioms of a probability distribution to show or answer the following:
\begin{enumerate}[label=(\alph*),topsep=0pt]
\item Show $\PP[\emptyset] = 0$.
\item Show $A\subseteq B \Longrightarrow \PP[A] \leq \PP[B]$.
\item Does $A\subsetneq B$ imply $\PP[A] < \PP[B]$?
\item Show $0\leq \PP[A] \leq 1$ for all $A$.
\end{enumerate}
\end{problem}
\begin{problem}
Use the axioms of a probability distribution to answer the following:
\begin{enumerate}[label=(\alph*),topsep=0pt]
\item Describe a situation where we might have a sample space $\Omega$ such that $\PP[\{\omega\}] = 0$ for all $\omega\in\Omega$.
\item
Explain what is wrong with the following proof that $1=0$.
\textbf{Proof.} By definition $\cup_{\omega \in \Omega} \{\omega\} = \{ \omega : \exists \omega' \in \Omega \text{ with } \omega \in \{\omega' \} \} = \Omega$.
Thus,
\begin{align*}
1 &= \PP[\Omega] \tag{Axiom 2}
\\&= \PP\bigg[\bigcup_{\omega \in \Omega} \{ \omega\}\bigg] \tag{definition of union}
\\&= \sum_{\omega \in \Omega} \PP[\{\omega\} ] \tag{Axiom 3}
\\&= \sum_{\omega \in \Omega} 0 \tag{assumption}
\\&= 0 \tag*{\qed}
\end{align*}
\end{enumerate}
\end{problem}
\begin{problem}[Wasserman 1.22 (simulate coin fipping)]
Suppose we flip a coin $n$ times and let $p$ denote
the probability of heads. Let $X$ be the number of heads. We call $X$
a binomial random variable, which is discussed in the next chapter.
Intuition suggests that $X$ will be close to $np$. To see if this is true, we
can repeat this experiment many times and average the $X$ values. Carry
out a simulation and compare the average of the $X$'s to $np$.
Try this for
$p =.3$ and $n = 10$, $n = 100$, and $n = 1000$ repeating each experiment 100 and then 1000 times.
Make a histogram of your results, or report the sample mean and variance.
\end{problem}
\begin{problem}[Wasserman 2.2 (computing probabilities from distributions)]
Let $X$ be such that $\PP[X=2] = \PP[X=3] = 1/10$ and $\PP[X=5] = 8/10$.
Plot the CDF $F$ for $X$ and use the CDF to find $\PP[2