I'm attaching the offending file as text, as I cannot send attachments
at the moment (is there a reasonable way to do this from
the command line? I can't use exmh right now because rcvstore is missing
from nmh for some reason, and it loses all my mail :(
Anyway, I reported this a couple of years ago, and it was fixed. Now
It's back. In the fourth line of the multiline equation in 1., something
has a double subscript. It's not visible in LyX, but it makes
LaTeX choke.
I'm assumint that it is the
p_{i}_{i}\mu
hawk
#LyX 1.2 created this file. For more info see http://www.lyx.org/
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\layout Title
Covariance--Key
\layout Author
Prof.
Hawkins
\layout Date
Due: February 13 MMI, 1:40 P.M.
\layout Standard
5 points.
Your result shouls be legible.
Turn in one for the entire group.
Include the entire derivation, not just the answers.
\layout Enumerate
\begin_inset LatexCommand \label{eqForVar}
\end_inset
Derive an easier formula for the variance of random variable
\begin_inset Formula \( X \)
\end_inset
, using
\begin_inset Formula \( \mu _{X} \)
\end_inset
.
\begin_deeper
\layout Standard
\begin_inset Formula \begin{eqnarray*}
\sigma _{X}^{2} & = & E\left[ \left( X-\mu _{X}\right) ^{2}\right] \\
& = & \sum _{i}p_{i}\left( X_{i}-\mu _{X}\right) ^{2}\\
& = & \sum p_{i}_{i}\left( X_{i}^{2}-2X_{i}\mu _{X}+\mu ^{2}_{X}\right) \\
& = & \sum _{i}p_{i}X_{i}^{2}-\sum _{i}p_{i}2X_{i}\mu _{X}+\sum p_{i}_{i}\mu
^{2}_{X}\\
& = & \sum p_{i}_{i}X_{i}^{2}-2\mu _{X}\sum p_{i}_{i}X_{i}+\mu ^{2}_{X}\sum
_{i}p_{i}\\
& = & \sum p_{i}_{i}X_{i}^{2}-2\mu _{X}\mu _{X}+\mu ^{2}_{X}1\\
& & \sum _{\iota }p_{i}X_{i}^{2}-\mu _{X}^{2}
\end{eqnarray*}
\end_inset
\end_deeper
\layout Enumerate
\begin_inset LatexCommand \label{values}
\end_inset
Let
\begin_inset Formula \( \mu _{X}=3 \)
\end_inset
,
\begin_inset Formula \( \sigma _{X}^{2}=16 \)
\end_inset
,
\begin_inset Formula \( \mu _{Y}=7 \)
\end_inset
and
\begin_inset Formula \( \sigma _{Y}^{2}=9 \)
\end_inset
.
Calculate the mean and variance of the following (assuming the independence
of
\begin_inset Formula \( X \)
\end_inset
and
\begin_inset Formula \( Y \)
\end_inset
:
\begin_deeper
\layout Enumerate
\begin_inset Formula \( 3X \)
\end_inset
\begin_deeper
\layout Standard
\begin_inset Formula \( \mu _{3X}=3\times 3=9 \)
\end_inset
\layout Standard
\begin_inset Formula \( \sigma _{3X}^{2}=3^{2}\times 16=144 \)
\end_inset
\end_deeper
\layout Enumerate
\begin_inset Formula \( 5Y \)
\end_inset
\begin_deeper
\layout Standard
\begin_inset Formula \( \mu _{5Y}=5\times 7=35 \)
\end_inset
\layout Standard
\begin_inset Formula \( \sigma _{5Y}^{2}=5^{2}\times 9=225 \)
\end_inset
\end_deeper
\layout Enumerate
\begin_inset Formula \( X+Y \)
\end_inset
\begin_deeper
\layout Standard
\begin_inset Formula \( \mu _{X+Y}=\mu _{X}+\mu _{Y}=3+7=10 \)
\end_inset
\newline
\begin_inset Formula \( \sigma ^{2}_{X+Y}=\sigma _{X}^{2}+\sigma _{Y}^{2}=16+9=25 \)
\end_inset
\end_deeper
\layout Enumerate
\begin_inset Formula \( 3X+2Y \)
\end_inset
\begin_deeper
\layout Standard
\begin_inset Formula \( \mu _{3X+2Y}=\mu _{3X}+\mu _{2Y}=3\mu _{X}+2\mu _{Y}=3\times
3+2\times 7=9+14=23 \)
\end_inset
\newline
\begin_inset Formula \( \sigma _{3X+2Y}^{2}=3^{2}\sigma _{X}^{2}+2^{2}\sigma
_{Y}^{2}=9\times 16+4\times 9=180 \)
\end_inset
\end_deeper
\end_deeper
\layout Enumerate
Calculate the covariance of
\begin_inset Formula \( aX \)
\end_inset
and
\begin_inset Formula \( bY \)
\end_inset
, where
\begin_inset Formula \( a \)
\end_inset
and
\begin_inset Formula \( b \)
\end_inset
are constants.
Your answer shouold be in terms of
\begin_inset Formula \( \sigma _{X} \)
\end_inset
,
\begin_inset Formula \( \sigma _{Y} \)
\end_inset
,
\begin_inset Formula \( \mu _{X} \)
\end_inset
,
\begin_inset Formula \( \mu _{Y} \)
\end_inset
\begin_inset Formula \( \sigma _{XY} \)
\end_inset
\begin_deeper
\layout Standard
\begin_inset Formula \begin{eqnarray*}
\sigma ^{2}_{aX+bY} & \equiv & E\left[ \left( \left\{ aX+bY\right\} -\mu
_{aX+bY}\right) ^{2}\right] \\
& = & E\left[ \left( \left\{ aX+bY\right\} -\mu _{aX}-\mu _{by}\right) ^{2}\right] \\
& = & E\left[ \left( \left\{ aX-a\mu _{X})\right\} +\left\{ bY-b\mu _{y}\right\}
\right) ^{2}\right] \\
& = & E\left[ \left( a\left\{ X-\mu _{X})\right\} +b\left\{ Y-\mu _{y}\right\}
\right) ^{2}\right]
\end{eqnarray*}
\end_inset
Then let
\begin_inset Formula \begin{eqnarray*}
A & \equiv & a\left\{ X-\mu _{X}\right\} \\
B & = & b\left\{ Y-\mu _{Y}\right\}
\end{eqnarray*}
\end_inset
continuing,
\begin_inset Formula \begin{eqnarray*}
\sigma _{aX+bY}^{2} & = & E\left[ \left( A+B\right) ^{2}\right] \\
& = & E\left[ A^{2}+2AB+B^{2}\right] \\
& = & E\left[ A^{2}\right] +E\left[ 2AB\right] +E\left[ B^{2}\right] \\
& = & E\left[ a^{2}\left( X-\mu _{X}\right) ^{2}\right] +2abE\left[ \left( X-\mu
_{x}\right) \left( Y-\mu _{y}\right) \right] +E\left[ b^{2}\left( Y-\mu _{y}\right)
^{2}\right] \\
& = & a^{2}\sigma _{X}^{2}+2ab\sigma _{XY}+b^{2}\mu _{Y}^{2}
\end{eqnarray*}
\end_inset
\end_deeper
\layout Enumerate
For the values in (
\begin_inset LatexCommand \ref{values}
\end_inset
), and the additional information that
\begin_inset Formula \( \sigma _{XY}=1 \)
\end_inset
calculate
\begin_deeper
\layout Enumerate
\begin_inset Formula \( 3X+2Y \)
\end_inset
\begin_deeper
\layout Standard
\begin_inset Formula \( \mu _{3X+2Y}=3\mu _{X}+2\mu _{Y}=3\times 3+2\times 7=23 \)
\end_inset
(unchanged)
\layout Standard
\begin_inset Formula \( \sigma _{3X+2Y}^{2}=3^{2}\sigma _{X}^{2}+2\times 3\times
2\times \sigma _{XY}+2^{2}\sigma _{Y}^{2}=9\times 16+12+4\times 9=192 \)
\end_inset
\end_deeper
\end_deeper
\layout Subsection*
Bonus
\layout Standard
Can you come up with a formula similar to the one in (
\begin_inset LatexCommand \ref{eqForVar}
\end_inset
) for covariance? If so, what is it? If not, why not?
\the_end
