Merge branch 'feature-robtillman'

Author: dwcoder

src/QuantitativePrimer.pdf

@@ -5,10 +5,10 @@
 % Should be of the format V<A>.<B>.<C>
 % A: Increment with major changes in layouts, large new sections added.
 % B: Increment when new questions are added
-% C: Increment with minor error corrections.
+% C: Increment with minor error corrections and additions
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-\newcommand{\docversion}{V1.1.1}
+\newcommand{\docversion}{V1.1.2}
 \begin{document}
 
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@@ -67,7 +67,11 @@
 
 \vfill
 \noindent
-Special thanks to Luke Miller for some clever solutions, William Xing for catching typos and suggestions on soft questions,  Iztok Pizorn for corrections in the code, Nita Bester for proofreading and support, and Joshua Curk for editing.
+Special thanks to Luke Miller and Robert Tillman for some clever solutions,
+William Xing for catching typos and suggestions on soft questions,
+Iztok Pizorn for corrections in the code,
+Nita Bester for proofreading and support,
+and Joshua Curk for editing.
 
 
 \clearpage

src/QuantitativePrimer.tex

@@ -217,6 +217,37 @@
 the third moment is related to the skewness,
 the second moment to the variance, and the first moment is the mean.
 
+Robert Tillman suggested the following clever solution, which is the shortest method to derive $E\left( X^4 \right)$ for the given normal distribution.
+Recognise
+\begin{align*}
+\Var(X^2) &= \E\left((X^2)^2\right) - \E\left( X^2 \right)^2  \\
+          &= \E\left(X^4\right) - \E\left( X^2 \right)^2  \\
+\E\left(X^4\right)
+&=  \Var(X^2) + \E\left( X^2 \right)^2
+\end{align*}
+and that
+\begin{align*}
+\frac{X^2}{\sigma^2} \sim \chi^2_{1}
+\text{.}
+\end{align*}
+The $\chi^2_k$ distribution has a mean of $k$ and variance of $2k$.
+Define a random variable
+$Z \sim \chi^2_{1}$
+and write
+\begin{align*}
+X^2 &= {\sigma^2} Z \\
+\E\left(X^2 \right) &= {\sigma^2} \E( Z ) \\
+                    &= {\sigma^2} (1) \\
+\Var\left(X^2 \right) &= (\sigma^2)^2 \Var( Z ) \\
+                    &= {\sigma^4} (2) \\
+\end{align*}
+giving the same answer as above,
+\begin{align*}
+\E\left(X^4\right) &= 2 \sigma^4 + \sigma^4 \\
+                   &= 3 \sigma^4
+\text{.}
+\end{align*}
+
 
 In my experience, interviewers from investment banks love questions about the normal distribution.
 This might be in response to interview solution manuals like