Moments and Stability of the Standard Normal Distribution

Let $\mathbb{R}$ be the set of real numbers and $\mathbb{N}$ the set of natural numbers.

Claim 1. Let $Z$ be a standard normal random variable on a probability space. Then $Z$, $Z^{2}$, and $|Z|^{3}$ are integrable, and the expectation and variance satisfy

$$\mathbb{E}[Z]=0,\qquad \mathbb{E}[Z^{2}]=\operatorname{Var}(Z)=1.$$

Claim 2. If $Z_1$ and $Z_2$ are 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.

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 square root of $n$.