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\begin{document}

\section*{Problem 1}
Reproduce the formatting of the following equation. You might want to use the following commands:  
\begin{verbatim}
\align;  \intertext;  \left; \right;  \sum; \prod \dots; \frac; \log; 
\end{verbatim}






\section*{Problem 2}
Suppose a crime has been committed. Blood is found at the scene for which there is
no innocent explanation. It is of a type which is present in $1\%$ of the population. 

\begin{enumerate}
    \item  The prosecutor claims: “There is a $1\%$  chance that the defendant would have the crime blood type if he were innocent. Thus there is a $99\%$ chance that he guilty”. This is known as the prosecutor’s fallacy.
    What is wrong with this argument?
 
   \item The defender claims: ``The crime occurred in a city of 800,000 people. The blood type would be
found in approximately 8000 people. The evidence has provided a probability of just 1 in 8000 that
the defendant is guilty, and thus has absolutely \textbf{no} bearing on the investigation.'' This is known as the defender’s fallacy. What is
wrong with this argument claiming the evidence has absolutely no bearing on the investigation?
\end{enumerate}

\section*{Problem 3}
My neighbor has two children. Assume that the gender of a child is a coin flip. 
Let the genders of the children be $G_1$ and $G_2$. For both questions, write the probability symbolically (e.g. ``$P(A|B)$'') and give the value.
\begin{enumerate}
    \item Suppose I happen to see one of his children run by, and it is a boy. What is the probability that the other child is a girl?
    \item Suppose instead that I ask him whether he has any boys, and he says yes. What is the probability that one child is a girl? 
\end{enumerate}


\section*{Problem 4}
Consider the Numbers game with one-sided interval hypotheses $h_{\leq x}$ for numbers $1$ up to $10$, where $h_{\leq x} = \{1,2,\dots, x \}$, $x\in\{1 \dots 10  \}$. Assume we have a uniform prior over $h$.
\begin{enumerate}
    \item Show that the $h_{mle} = h_{\leq \text{max}(S)}$ for a given set of numbers $S = \{x_1, x_2, \dots\}$.

    \item  Briefly say why the MAP estimate = MLE.
    
    \item For Parts 3 and 4, let's suppose $S =\{5,9\}$. What is the plug-in approximation of the posterior predictive distribution for new data point $x$?
    
    \item What is the full posterior predictive distribution? 
    
  
\end{enumerate}





\end{document}