add solution to problem 5.21

Author: RamonAraujo

src/chapters/5/sections/normal/index.tex

@@ -1,5 +1,7 @@
 \section{Normal}
 
+\subsection{problem 21}
+\input{problems/21}
 \subsection{problem 26}
 \input{problems/26}
 \subsection{problem 35}

src/chapters/5/sections/normal/problems/21.tex

@@ -0,0 +1,45 @@
+(a)
+Let's begin calculating the second derivative of the standard Normal PDF
+
+$$
+\Phi(z) = \frac{1}{\sqrt{2\pi}} \, e^{-z^2/2}
+$$
+
+$$
+\Phi^{'}(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2} \frac{(-2z)}{2} = -\frac{z}{\sqrt{2\pi}} \, e^{-z^2/2}
+$$
+
+\begin{equation*}
+\begin{split}
+\Phi^{''}(z)
+&= -\frac{1}{\sqrt{2\pi}} \left[ e^{-z^2/2} + z e^{-z^2/2} \frac{(-2z)}{2} \right] \\
+&= \frac{1}{\sqrt{2\pi}} e^{-z^2/2} (z^2 - 1)
+\end{split}
+\end{equation*}
+
+The second derivative of the PDF is zero at the points of inflection $z_\mathrm{inf}$.
+
+$$
+\Phi^{''}(z_\mathrm{inf}) = 0
+$$
+
+$$
+z_\mathrm{inf}^2-1 = 0
+$$
+
+$$
+z_\mathrm{inf} = \pm 1
+$$
+
+The points of inflection of the standard Normal PDF are at $-1$ and $+1$.
+
+(b)
+Let $Z \sim \mathcal{N}(0,1)$ and $X \sim \mathcal{N}(\mu,\sigma^2)$.
+
+We can relate $X$ to $Z$ by the location-scale transformation
+
+$$
+X = \mu + \sigma Z
+$$
+
+Substituting $Z = \pm 1$ into the above relation, we have that the points of inflection of the $\mathcal{N}(\mu,\sigma^2)$ PDF are at $(\mu-\sigma)$ and $(\mu+\sigma)$.