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\begin{document}
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\begin{flushright}
Name \underline{\hspace{60mm}} \\ 
$\,$ \\
Student Number \underline{\hspace{60mm}}
\end{flushright}   
\vspace{2mm}

\begin{center}   
{\Large \textbf{STA 312 f2023 Quiz 2}}\\
\vspace{1 mm}
\end{center}   

\begin{enumerate}

\item (5 points) Let $X_1, \ldots, X_n$ be a random sample from a geometric distribution; see formula sheet. Find the maximum likelihood estimate of $\theta$. Show your work and \textbf{circle your final answer}. \emph{You do not need to bother with the second derivative test.} 

\pagebreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item \label{comp} For Question 9 of Assignment 2, you analyzed numerical data from a distribution with density $f(x|\pi) = \pi e^{-\pi/x} \frac{1}{x^2}$ for $x>0$, where the parameter $\pi>0$. 
    \begin{enumerate}
        \item (2 points) \label{mle} In the space below, write $\widehat{\pi}$, the maximum likelihood estimate of $\pi$. The answer is a number from your printout. On your printout, circle the number and write ``Question~\ref{mle}" beside it. \vspace{80mm}
        \item (3 points) \label{ci} You found a 95\% confidence interval for $\pi$. In the space below, write the lower and upper confidence limits. The answer is a pair of numbers from your printout. On your printout, circle the numbers and write ``Question~\ref{ci}" beside them. 
    \end{enumerate}


\end{enumerate}\vspace{80mm}

\noindent
Please attach the printout with your answers to Question~\ref{comp} of this quiz (Question 9 of the assignment). Make sure your name and student number are written on the printout.

\end{document}


# Assignment 2
# Simulating inverse gamma (one over exponential) data with lambda = 3. 
# Name of the file will be inversegamma.data.txt.

rm(list=ls()); options(scipen=999)
n = 300; lambda=3
set.seed(9999)
x = 1/rexp(n,rate=lambda); x = round(x,2); x 

    6.56    1.47    7.92    4.35    1.54    6.23    4.71    0.73    1.74    5.66    2.25
    1.28    4.44    1.98    3.00   12.79    2.37    2.92    1.40    3.22   13.71    4.28
    1.28    6.29   17.05   24.18    0.94    2.91    9.08    3.13   57.68   12.75   57.86
    1.03    0.82   12.71    8.67   46.58   11.07    5.28    4.28    8.40    1.26    3.33
    1.46   14.98    7.65    1.22   26.03   14.91    1.04   21.10    3.30    3.25    3.81
    1.97    4.71    4.56    2.63   14.60    4.72   15.42    2.98    0.66    2.20   11.10
   14.92    1.46    2.37   11.00  325.18    1.99    0.78    1.26    2.74    4.29    3.60
    2.16    3.98    1.68    7.03    2.46    2.97    1.65    3.07  585.02    3.65   11.13
    2.36   26.05    1.51 1536.99   84.13    2.50   23.69    4.60    4.11    8.49   72.78
    4.03   13.56    2.98    1.36    9.99    0.53    4.93    4.80    2.84    1.14    1.53
    1.38    5.95    2.31    3.93   38.61    9.28    1.05    4.42   17.52   13.29    1.41
    2.40   28.88    2.73    3.32   35.54   13.25    8.18    1.84    6.60    2.54    1.86
   17.16    5.46    1.34    3.49   33.79    2.08    2.17    9.31   50.09    2.39    6.14
    2.60    1.60    1.89    4.46    8.08    1.17    1.04    1.05    2.34   13.01   11.29
    4.31    2.15   30.32    1.67   32.96    3.10    1.70    1.80    1.01    8.85    2.22
    1.03    3.46    0.54 5096.32    3.09   14.56   11.13   36.13   10.05    4.02    2.66
   25.56   12.38    8.55   10.19    5.29    1.57    8.66    8.94    2.43   19.45   16.15
    0.94  239.27    2.96    4.84    4.52   11.99    1.26    2.52    2.78    1.72    2.43
    1.74    8.89    1.37    1.63    6.26    2.55    3.83    3.76    1.58    2.86    1.38
    8.38  215.87   27.51    3.47    1.73    3.89    1.70   31.09    9.19   11.51    1.04
    1.27    0.42    3.68   13.26   16.91    6.01    4.18   22.11    4.58    4.31   14.92
    8.70    4.89    1.11    7.27    0.93    4.84    5.36   15.09    0.73    1.63    2.14
  145.46    1.00    3.09  103.53    2.40    2.54   17.21    2.08   14.38    1.44    1.81
    5.19    7.28   29.13    2.40    6.65   12.05    6.71    3.54    1.44    2.87    1.44
    1.39   20.34    0.77    5.65    0.83    1.77    9.61    4.74   26.91    1.97    3.85
    1.77    1.42    1.84   22.22    8.02    5.13  144.21    2.55    3.94    2.45    1.92
    2.41    1.14    7.47    1.20   59.21    2.18   25.70    1.34    3.08    0.95    7.15
    5.96    2.32    3.96

x = scan("http://www.utstat.toronto.edu/~brunner/data/legal/inversegamma.data.txt")
n = length(x)
pihat = 1/mean(1/x); pihat
se_pihat = sqrt(pihat^2/n); se_pihat
Z = (pihat-3.14159)/se_pihat; Z
pvalue = 2 * (1-pnorm(abs(Z))); pvalue


> x = scan("http://www.utstat.toronto.edu/~brunner/data/legal/inversegamma.data.txt")
Read 300 items
> n = length(x)
> pihat = 1/mean(1/x); pihat
[1] 2.740541
> se_pihat = sqrt(pihat^2/n); se_pihat
[1] 0.1582252
> Z = (pihat-3.14159)/se_pihat; Z
[1] -2.534672
> pvalue = 2 * (1-pnorm(abs(Z))); pvalue
[1] 0.01125527
> 
>