The standard normal distribution is a special case of the [[normal distribution]] and is often used as a reference distribution in statistics. Use a [lookup table](https://www.math.arizona.edu/~rsims/ma464/standardnormaltable.pdf) or the `qnorm` function in R for reference values. The expected value and variance are the same as the general form of the normal distribution. ## Notation $Z \sim N(0,1)$ The notation $Z$ and $\Phi$ are often associated with the standard normal distribution. ## Probability Density Function $f_Z(x) = \frac{1}{\sqrt{2\pi}} \ e^{-x^2 / 2}$ ## Cumulative Density Function The cdf is so important in statistics it gets its own name $\Phi$. $F(Z) = \Phi(z) = P(Z <= z) = \int\limits_{-\infty}^{z}\frac{1}{\sqrt{2\pi}}e^{-x^2/2}dx$ The cdf cannot be expressed in closed form. Historically, statisticians looked up the [[numerical analysis|numerically]] calculated values in look up tables, but now we use [[R]].