Let $\mathbb{R}$ be the set of \reftext{def:real-numbers-c54-2026c}{real numbers} and $\mathbb{N}$ the set of \reftext{def:natural-numbers-2026a}{natural numbers}.

\textbf{Claim 1.} Let $Z$ be a \reftext{def:standard-normal-distribution-2026a}{standard normal} random variable on a \reftext{def:probability-space-random-variable-2026a}{probability space}. Then $Z$, $Z^{2}$, and $|Z|^{3}$ are \reftext{def:lebesgue-integral-integrable-2026a}{integrable}, and the \reftext{def:expectation-variance-2026a}{expectation and variance} satisfy

\textbf{Claim 2.} If $Z_1$ and $Z_2$ are \reftext{def:independence-events-rvs-2026a}{independent} standard normal random variables on a common probability space and $a,b$ are positive real numbers with $a^{2}+b^{2}=1$, then $aZ_1+bZ_2$ is a standard normal random variable.

\textbf{Claim 3.} If $n\in\mathbb{N}$ with $n\ge 1$ and $Z_1,\dots,Z_n$ are independent standard normal random variables on a common probability space, then $(Z_1+\cdots+Z_n)/\sqrt{n}$ is a standard normal random variable, where $\sqrt{n}$ denotes the positive \reftext{thm:nonnegative-real-has-unique-square-root-2026a}{square root} of $n$.