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Name \underline{\hspace{60mm}} \\ 
$\,$ \\
Student Number \underline{\hspace{60mm}}
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{\Large \textbf{STA 312 s2023 Quiz 6}}\\
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\begin{enumerate}

\item The Kapan-Meier estimate of the survival function is based on discrete time. Accordingly, let the survival time $T$ be a discrete random variable with non-zero probability on the points $t_1, t_2, \dots$. Let $p_j = $ the probability of surviving past time $t_j$, given survival to time $t_{j-1}$. That is, $p_j = P(T>t_j|T>t_{j-1})$. 
    \begin{enumerate}
        \item (3 points) Prove $p_j = \frac{S(t_j)}{S(t_{j-1})}$. \vspace{80mm}
        \item (3 points) Assuming  $t_0=0$ and $P(T>0)=1$ (which is very reasonable), prove $S(t_3) = p_1 \, p_2 \, p_3$.  This is a special case of something on the formula sheet, and of course you cannot use what you are proving.
    \end{enumerate}

 \vspace{75mm}

\item (4 points) \label{comp} In Question 12 of Assignment 6, you were asked to plot the Kaplan-Meir and maximum likelihood estimates of $S(t)$ for a small data set. Attach the plot and the R code that produced it to this quiz. On your printout, mark \emph{the code that added the maximum likelihood estimate of $S(t)$} the plot, and and write ``Question~\ref{comp}" beside it. Make sure your name and student number appear on the printout and plot.


\end{enumerate}

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