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Prof.~David Draper \\
Department of Applied Mathematics and Statistics \\
University of California, Santa Cruz

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\textbf{\large AMS 131: Quiz 6}

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Name: \underline{\hspace*{5.85in}}

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(Note that part (e) of this question is on the second page.)

In a problem you're working on, you need to simulate random draws from the following PDF for the continuous random variable $Y$:
\begin{equation} \label{y-pdf-1}
f_Y ( y ) = \left\{ \begin{array}{cc} \frac{ 1 }{ 2 } ( 2 \, y + 1 ) & \textrm{for } 0 \le y \le 1 \\ 0 & \textrm{otherwise} \end{array} \right\} \, .
\end{equation}

\begin{itemize}

\item[(a)]

Sketch the PDF in equation (\ref{y-pdf-1}) for $y$ in the interesting range $[ 0, 1 ]$.

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\item[(b)]

Work out the CDF $F_Y ( y )$ for $Y$, specifying its values for all $- \infty < y < + \infty$, and sketch it in the interesting range $0 \le y \le 1$.

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\item[(c)]

Work out the inverse CDF (quantile function) $F_Y^{ -1 } ( p )$ for $Y$, specifying its values for all $0 < p < 1$, and sketch it for $p$ in that range.

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\item[(d)]

Building on your result in part (c), explicitly specify how you can generate IID random draws from the PDF in equation (\ref{y-pdf-1}).

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\item[(e)]

Once you have your random sample in part (d), briefly explain how you could graphically check whether it really \textit{is} a sample from the PDF in equation (\ref{y-pdf-1}).

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