Standard normal distribution ($N(0,1)$)

The standard normal distribution is $N(0,1)$ and has cumulative distribution function $\Phi$.

The standard normal density is

Completing the square in the standard normal density gives $\mathbb{E}[e^{tZ}]=e^{t^2/2}$. Differentiating at zero yields $\mathbb{E}[Z]=0$, $\mathbb{E}[Z^2]=1$, and $\mathbb{E}[Z^4]=3$.

For $Z$ with a standard normal distribution and $a\ge 0$, split the Gaussian integral at zero and complete the square to obtain $\mathbb{E}{e}^{a|Z|}=2e^{a^2/2}\Phi (a)\le 2e^{a^2/2}$, where $\Phi$ is the standard normal distribution function. This converts a bound on an absolute Gaussian random variable into a two-sided Gaussian tail bound using Markov inequality.

The standard normal distribution function is

The standard normal quantile $z_q$ is the unique real number satisfying $\mathbb{P}(Z\le z_q)=q$ for $Z\sim N(0,1)$ and $0<q<1$.

If $X\sim N(\mu ,{\sigma }^2)$ with $\sigma >0$, then $(X-\mu )/\sigma$ has the standard normal distribution.