$\DeclareMathOperator{\p}{Pr}$ $\DeclareMathOperator{\P}{Pr}$ $\DeclareMathOperator{\c}{^C}$ $\DeclareMathOperator{\or}{ or}$ $\DeclareMathOperator{\and}{ and}$ $\DeclareMathOperator{\var}{Var}$ $\DeclareMathOperator{\E}{E}$ $\DeclareMathOperator{\std}{Std}$ $\DeclareMathOperator{\Ber}{Bern}$ $\DeclareMathOperator{\Bin}{Bin}$ $\DeclareMathOperator{\Poi}{Poi}$ $\DeclareMathOperator{\Uni}{Uni}$ $\DeclareMathOperator{\Exp}{Exp}$ $\DeclareMathOperator{\N}{N}$ $\DeclareMathOperator{\R}{\mathbb{R}}$ $\newcommand{\d}{\, d}$

Normal Cumulative Density Function

To calculate the Cumulative Density Function (CDF) for a Normal random variable at a value $x$, also writen as $F(x)$, you can transform your distribution to the "Standard Normal" and look up the corresponding value in the Standard Normal CDF. However, most programming libraries will provide a Normal cdf function. This tool replicates said functionality.



This function calculates the cumulative density function of a Normal random variable. It is very important in CS109 to understand the difference between a probability density function (PDF), and a cumulative density function (CDF). The CDF of a random variable at point little $x$ is equal to the probability that the random variable takes on a value less than or equal to $x$. If the random variable is called big $X$, the CDF can be written as $P(X < x)$ or as $F_X(x)$.

The CDF function of a Normal is calculated by translating the random variable to the Standard Normal, and then looking up a value from the precalculated "Phi" function ($\Phi$), which is the cumulative density function of the Standard Normal. The Standard Normal, often written $Z$, is a Normal with mean 0 and variance 1. Thus, $Z \sim N(\mu = 0, \sigma^2 = 1)$.

Try different calculations to see different translations to the Standard Normal!

Standard Normal PDF

The Normal transform means that the probability that you are calculating is identical to the integral over the PDF of the Standard Normal from $-\infty$ to $z=(x-\mu)/\sigma$. This picture shows that integral: