Topics: Probability - Distribution Function


Let be an absolutely continuous random variable.

We say that has a normal distribution (or Gaussian distribution) if its density function is:

…and write .

In particular, the random variable such that is of special importance: its distribution is called the standard normal distribution.

Expected Value and Variance

Notice that this probability distribution depends on two parameters: and .


The expected value of an with a normal distribution is simply and elegantly:


The variance of an with a normal distribution is simply and elegantly:

Indeed, we use these two values as parameters since:

  • provides the value around which most of the mass concentrates. It is a localisation parameter: if we change it, the curve moves horizontally, but it stays otherwise the same.

  • provides the distribution’s standard deviation (i.e. a measure of how disperse the mass is around ). It is a dispersion parameter: if we change it, the concentration of the mass changes, but the curve stays in the same location.


A few examples of the density distribution graphs of various normal distributions with different parameters:

Notice how, effectively, the mass concentrates around and provides a dispersion measure, as mentioned before.

The red curve is the standard normal distribution.