Location-scale normal laws have their stated mean and variance

Given: AC, a standard normal random variable $Z$, and $X$ with law $N(m,{\sigma }^2)$ for $m\in \mathbb{R}$ and $\sigma \ge 0$.

AC is required by the standard normal probability-law definition and the Gaussian even-moment supplier. The Axiom of Choice Standard normal and normal laws Gaussian even moments for Brownian increments

$N(0,1)$ is the law with density $\varphi (x)=e^{-x^2\mathrm{/}2}\mathrm{/}\sqrt{2\pi }$, and $N(m,{\sigma }^2)$ is the law of $x\mapsto m+\sigma x$ under $N(0,1)$. Standard normal and normal laws

For a standard normal $Z$, $E[Z^2]=1$. Gaussian even moments for Brownian increments

Cauchy–Schwarz gives $E\mathrm{\mid }Z\mathrm{\mid }\le (E[Z^2]{)}^{1\mathrm{/}2}$, so the standard normal is integrable. Cauchy-Schwarz for random variables

A nonnegative measurable function of a random element has expectation equal to its integral against the law, allowing infinity; the same identity holds for a signed function once its composite is integrable. Change of variables for expectation

Expectation is linear, and variance is $E[(X-E[X]{)}^2]$ for square-integrable $X$. Linearity, monotonicity, and the modulus bound for expectation Moments, variance, and covariance on a probability space

A pointwise-convergent sequence dominated by one integrable function has convergent integrals. Dominated convergence

The substitution $u=-x$ is valid in the proper integral of the continuous normal density on a compact interval. Substitution: if $\phi$ is differentiable on $[c,d]$ with ${\phi }^{\mathrm{\prime }}$ integrable and $f$ is continuous on an interval containing $\phi ([c,d])$, then ${\int }_{\phi (c)}^{\phi (d)}f={\int }_c^d(f\circ \phi ){\phi }^{\mathrm{\prime }}$