Q2
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└── Homework
└── W7
└── Q2.tex
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\fancyhead[l]{Li Yifeng}
\fancyhead[c]{Homework \#7}
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\begin{document}
\section*{Question 2}
\noindent Assume that the random variable $X$ has the following probability mass function:
\[
p(x) =
\begin{cases}
\begin{aligned}
\frac{1}{8},&\quad x = -1\\
\frac{1}{4},&\quad x = 0\\
\frac{1}{2},&\quad x = 1\\
\frac{1}{8},&\quad x = 2
\end{aligned}
\end{cases}
\]
\bigskip
\begin{enumerate}[start=1,label={\bfseries Part \arabic*:},leftmargin=0in]
\bigskip\item Compute the expectation of $X$ and of $X^2$.
\subsection*{Solution}
Easy to get
\[
\begin{aligned}
\bE[X] &= -1p(-1) + 0p(0) + 1p(1) + 2p(2)\\
&= -\frac{1}{8} + 0 + \frac{1}{2} + \frac{1}{4}\\
&= \frac{5}{8}
\end{aligned}
\]
As well as
\[
\begin{aligned}
\bE\left[X^2\right] &= (-1)^2p(-1) + 0^2p(0) + 1^2p(1) + 2^2p(2)\\
&= \frac{1}{8} + 0 + \frac{1}{2} + \frac{1}{2}\\
&= \frac{9}{8}
\end{aligned}
\]
\subsection*{Answer}
\[\boxed{\begin{aligned}
\bE[X] &= \frac{5}{8}\\
\bE\left[X^2\right] &= \frac{9}{8}
\end{aligned}}\]
\bigskip\item Compute the variance of $X$ and of $X^2$.
\subsection*{Solution}
Easy to get the variance of $X$
\[
\begin{aligned}
V(X) &= \bE\left[X^2\right] - (\bE[X])^2\\
&= \frac{9}{8} - \left(\frac{5}{8}\right)^2\\
&= \frac{47}{64}
\end{aligned}
\]
As well as the variance of $X^2$
\[
\begin{aligned}
V(X^2) &= - \left(\bE\left[X^2\right]\right)^2 + \bE\left[\left(X^2\right)^2\right]\\
&= -\frac{81}{64} + \left((-1)^4p(-1) + 0^4p(0) + 1^4p(1) + 2^4p(2)\right)\\
&= -\frac{81}{64} + \left(\frac{1}{8} + 0 + \frac{1}{2} + 2\right)\\
&= \frac{87}{64}
\end{aligned}
\]
\subsection*{Answer}
\[\boxed{\begin{aligned}
V(X) &= \frac{47}{64}\\
V(X^2) &= \frac{87}{64}
\end{aligned}}\]
\end{enumerate}
\end{document}