#### Definition 2.2.8.

Consider the parametrized family of functions:
\begin{gather*} f(x; \mu, \sigma^2) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{\frac{-(x-\mu)^2}{2\sigma^2}}. \end{gather*}
The function \(f(x; \mu, \sigma^2)\) is called a normal density function with parameters \(\mu\) and \(\sigma^2\text{.}\) A continuous random variable \(X\) with pdf \(f(x; \mu, \sigma^2)\) is said to have a normal distribution. We’ll write \(X \sim \Norm(\mu, \sigma^2)\text{.}\)
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In the specific case that \(\mu = 0\) and \(\sigma^2 = 1\text{,}\) we call the resulting distribution the standard normal distribution. We’ll use the notation \(\phi(x)\) for the standard normal density function, and \(\Phi(x)\) for the standard normal cumulative distribution function.
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