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## Background

Gaussian sampling — that is, generating samples from a Gaussian distribution — plays an important role in many cutting-edge fields of data science, such as Gaussian process, variational autoencoder, or generative adversarial network. As a result, you often see functions like tf.random.normal in their tutorials.

But, deep down, how does a computer know how to generate Gaussian samples? This series of blog posts will show 3 different ways that we can program our computer (via Python) to do so. You will also see how R and Python generate Gaussian samples using modified versions of some of these methods.

## Starting point: the uniform number generator

Of course, we can’t generate Gaussian samples from thin air. Instead, we start with a random number generator that exists in almost all programming languages: the **uniform random number generator**. It generates a random number that could take any value between 0 and 1. For Python, the numpy.random module uses the Mersenne twister to generate a uniformly-distributed float that is in the interval [0.0, 1.0).

Since Gaussians are better visualized in 2 dimensions — we are all familiar with the Gaussian “blob” in the xy-plane — I will demonstrate the 3 sampling methods in 2-D, especially since one of the methods do in fact generate Gaussians in two dimensions at the same time.

As a result, this series is broken down into 3 parts (see accompanying image):

- The first two part — part 1 and part 2 — describes 2 common methods to transform the uniform samples (in gray) into points in which their x- and y-coordinates are independent standard Gaussians: both coordinates have mean of 0, variance of 1, and their covariance is 0 (in red). The first method, the
**inverse transform sampling**, is described below. - The last part — part 3 — reveals a much simpler alternative to the the two methods above in generating standard Gaussian samples. It will also show how to transform these 2-D standard Gaussian samples (in red) to have any given mean or variance in x and y, as well as any given covariance between the x and y coordinates (in blue).

## Method 1: Inverse transform sampling

This is the most basic, yet most common, way to convert a uniform random sample into a random sample of any distribution, including Gaussian. This method works by applying the **inverse function of the Gaussian CDF** (cumulative distribution function) to transform a uniform sample to a Gaussian sample.

To make sure that the Gaussian samples for the x- and y-coordinates are independent, we can use two different uniform samples, one for x (U₁), and one for y (U₂). These two uniform samples can be generated using two different random number generators (two different `RandomState`

initialized by different seeds, for example) so that they are independent in the first place.

## How does this work?

This method works by exploiting a mind-blowing principle:

For any distribution, the cumulative probability is always uniformly distributed.

The arithmetic proof of this principle is straight-forward but rather boring, and you can view it from this lecture. Instead, I will show the geometric interpretation of this principle. But first, let’s clarify what cumulative probability is: