Standard Normal Distribution

Define $g:\mathbb{R}\to\mathbb{R}$ by $g(x)=\exp(-x^{2}/2)$, with the exponential function; $g$ is continuous (a composition of continuous maps, by Composition of Continuous Euclidean Maps and claim 3 of Basic Properties of the Exponential Function), hence measurable with respect to the Borel $\sigma$-algebra by the generator criterion there (preimages of open sets are open).

By claim 2 of The Gaussian Weight Defines a Probability Distribution, the quantity

(the integral of Lebesgue Integral of a Nonnegative Measurable Function with respect to Lebesgue measure $\lambda$) is a finite positive real number. The standard normal distribution is the function

with $\mathbf{1}_B$ the indicator of Simple Function and Its Integral. By claims 1 and 2 of The Gaussian Weight Defines a Probability Distribution, $N$ is a probability measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. Its cumulative distribution function is denoted

and $\Phi$ is continuous at every point (claim 3 of The Gaussian Weight Defines a Probability Distribution). A random variable is called standard normal (or standard Gaussian) if its distribution is $N$.