# 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.

## Explanation

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: