Theorem 9.2.2. Standard Normal Distribution.
If \(\mu = 0\) and \(\sigma = 1\text{,}\) then we say X has a standard normal distribution and often use Z as the variable name and will use \(\Phi(z)\) for the standard normal distribution function. In this case, the density function reduces to
\begin{equation*}
f(z) = \frac{1}{\sqrt{2 \pi}} e^{ -z^2 / 2}
\end{equation*}