Q1
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Probability
└── Homework
└── W7
└── Q1.tex
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\fancyhead[l]{Li Yifeng}
\fancyhead[c]{Homework \#7}
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\begin{document}
\section*{Question 1}
\noindent Eight thousand lottery tickets are sold for \$5 each.
\bigskip
\noindent One ticket will win $\$2,000$, two tickets will win $\$750$ each, and five tickets will win $\$100$ each.
\bigskip
\begin{enumerate}[start=1,label={\bfseries Part \arabic*:},leftmargin=0in]
\bigskip\item Let $X$ denote the net gain from the purchase of a randomly selected ticket. Construct the probability distribution of $X$.
\subsection*{Solution}
Easy to get the $pdf$ of $X$
\[
p(x) =
\begin{cases}
\begin{aligned}
\frac{1}{8\,000},&\quad x = 2\,000 - 5 &= 1\,995\\
\frac{2}{8\,000},&\quad x = 750 - 5 &= 745\\
\frac{5}{8\,000},&\quad x = 100 - 5 &= 95\\
\frac{8\,000 - 1 - 2 - 5}{8\,000} = \frac{7\,992}{8000},&\quad x = 0 - 5 &= -5
\end{aligned}
\end{cases}
\]
\subsection*{Answer}
\[\boxed{p(x) =
\begin{cases}
\begin{aligned}
\frac{1}{8\,000},&\quad x = 1\,995\\
\frac{2}{8\,000},&\quad x = 745\\
\frac{5}{8\,000},&\quad x = 95\\
\frac{7\,992}{8000},&\quad x = -5
\end{aligned}
\end{cases}}\]
\bigskip\item Compute the expected value $\mu$ of $X$ and intepret its meaning.
\subsection*{Solution}
As given, easy to get
\[
\begin{aligned}
\mu &= 1\,995p(1\,995) + 745p(745) + 95p(95) + -5p(-5)\\
&= \frac{399}{1\,600} + \frac{149}{800} + \frac{19}{320} - \frac{999}{200}\\
&= -\frac{9}{2}
\end{aligned}
\]
\subsection*{Answer}
\[\boxed{\mu = -\frac{9}{2}}\]
\bigskip\item Compute the standard deviation $\sigma$ of $X$.
\subsection*{Solution}
Easy to get
\[
\begin{aligned}
\sigma &= \sqrt{-\mu^2 + \bE\left[X^2\right]}\\
&= \sqrt{-\mu^2 + \left(1\,995^2p(1\,995) + 745^2p(745) + 95^2p(95) + (-5)^2p(-5)\right)}\\
&= \sqrt{-\frac{81}{4} + \left(\frac{398\,0025 + 111\,0050 + 45\,125 + 199\,800}{8\,000}\right)}\\
&= \sqrt{\frac{5\,173}{8}}\\
&\approx 25.429
\end{aligned}
\]
\subsection*{Answer}
\[\boxed{\sigma = \sqrt{\frac{5\,173}{8}} \approx 25.429}\]
\end{enumerate}
\end{document}