Definition (Standard normal random variable)

A continuous RV $Z$ is called a standard normal if its pdf is $$f(x)=\frac{1}{\sqrt{2\pi}}e^{\frac{-x^{2}}{2}}$$with cdf $$\Phi(z)=P(Z<z)=\int_{-\infty}^{z}e^\frac{-t^{2}}{2}dt$$For RV $Z\sim N(0,1)$ $$\begin{align*} E[Z]&=0\\ \mbox{Var}(Z)&=1 \end{align*}$$