## 4.4 Distribution functions

For each distribution, there is more than one distribution function. When considering the Normal distribution so far, the type of distribution function we have been considering has been the Probability Density Function (PDF). Two other distribution functions are the Cumulative Distribution Function (CDF) and the quantile function. We will describe each function now, and then explain further via an example.

Probability Density Functions

For a continuous random variable $$X$$, the Probability Density Function (PDF) is a function that tells us the the density of probability at a given value. Or, more usefully, the area under the curve of a PDF tells us the probability of $$X$$ falling within a certain range of values.

Cumulative Distribution Functions

For a continuous random variable $$X$$, the Cumulative Distribution Function (CDF) is a function that tells us, for a given value $$x$$, the probability that $$X$$ is less than or equal to $$x$$. That is, the value of the function at $$x$$ is equal to $$P(X \leq x)$$.

Quantile Function

For a continuous random variable $$X$$, the quantile function tells us the value of $$x$$ for which the quantile would be equal to a certain value.

Let us again consider the continuous random variable $$X$$ that denotes the height in cm of university students and is normally distributed such that $$X∼N(172.38,9.852)$$. Let us also consider a height of $$x = 165$$ and see how it can be represented in each type of function below:

These three functions are related mathematically: For example, using calculus, if we know the PDF, we can derive the CDF by taking the integral of the PDF. It is also true that the quantile function is the inverse of the CDF. In this subject however, we will be using statistical software packages to help us navigate these functions as needed.