Standard Normal Distribution

In the setting of Probability Spaces: Notation and Product Measures, Lebesgue Measure on ℝⁿ and Image Measures: Notation, let $\lambda$ be Lebesgue measure on $\mathbb{R}$, let $\exp$ be the exponential function, and let

$$c_{\mathrm{G}}=\int_{\mathbb{R}}\exp(-x^{2}/2)\,d\lambda(x),$$

a positive real number by The Gaussian Integral Is Finite and Positive, and the First Two Gaussian Moments (i).

(i) Standard normal density. The standard normal density is the function $\varphi:\mathbb{R}\to\mathbb{R}$,

$$\varphi(x)=\frac{1}{c_{\mathrm{G}}}\exp(-x^{2}/2).$$

(ii) Standard normal distribution. The standard normal distribution is the measure $\gamma=\varphi\lambda$ on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ with density $\varphi$ with respect to $\lambda$ ($\varphi$ is continuous, since $x\mapsto-x^{2}/2$ is continuous by Continuity of Identity Maps, Compositions and Reciprocals (i) and Continuity of Sums and Products of Real-Valued Functions on a Metric Space (v), $\exp$ is continuous by Basic Properties of the Exponential Function (viii), and composites and constant multiples of continuous maps are continuous by Continuity of Identity Maps, Compositions and Reciprocals (ii) and Continuity of Sums and Products of Real-Valued Functions on a Metric Space (v); it is positive by Basic Properties of the Exponential Function (iii) and $c_{\mathrm{G}}>0$; hence it is an element of $\mathcal{M}^{+}(\mathbb{R},\mathcal{B}(\mathbb{R}))$, by Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iv) and Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (vii)):

$$\gamma(B)=\int_{B}\varphi\,d\lambda\qquad(B\in\mathcal{B}(\mathbb{R})).$$

(iii) Standard normal random variables. A random variable $Z$ is standard normal if its law is $\gamma$:

$$P_{Z}=\gamma.$$