Standard Normal Random Variable

A continuous random variable ${\cal N}(\mu,\sigma^2)$ is defined by a density function f:

$$f(z)= \phi(z)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}z^2}$$

Standard Normal Integrals:

$$\int_{-\infty}^{+\infty}\phi(z)=1 \qquad ; \qquad \int_{-\in... ...\phi(z)=0 \qquad; \qquad \int_{-\infty}^{+\infty}z^2\phi(z)=1$$

 

Important fact:

$$\Phi(x)=\int_{-\infty}^{x}\phi(z) \mbox{ is NOT available in closed form}$$