**Contents**

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In chapter 2, we have mentioned the normal (or gaussian) probability distribution. We now have the knowledge needed to introduce and understand its equation. Why studying the normal distribution? This distribution is very common in nature. For example the adult height in any adult given population generally follows (more or less) a normal distribution. We will see it reapparing in the next chapter as well.

Gaussian or normal distribution have a typical bell-shaped curve (see figure 1). The equation for this distribution is a bit complex:

$$p(x) = \mathcal{N}(\mu, \sigma) = {\dfrac{1}{\sigma \sqrt {2 \pi} } } e^{-{\dfrac{(x -\mu)^2}{2\sigma^2}}}.$$You should already be familiar with the \(\sigma\) term (the greek letter sigma) which is the distribution's **standard deviation** (see the previous chapter if you need a refresher on what the standard deviation is). The term \(\mu\) (the greek letter mu) is the distribution's expectation (a concept we have also introduced in the previous chapter). Note that the curve is symmetrical around \(\mu\). By changing the value of these parameters we can obtain all sort of bell-shaped distributions as pictured in figure 2. The normal distribution function is generally denoted \(\mathcal{N}(\mu, \sigma)\) since mean and variance are the two parameters of the model. \(\mathcal{N}(0,1)\) is called the standard normal distribution (figure 1).

In the next chapter, we will learn about the concepts of PDF (probability density function) which is derived from the concept of probability distribution, and CDF (cumulative distribution function). These two concepts are very important in computer graphics.