The statistical (or probability) distribution is a parameterized mathematical function which can be used to compute the probability of any single observation from the sample space. This function defines the density of the observations, known as probability density function (PDF). The latter describes how the probability of the proportion of observations change over the range of the distribution. More in practice, a distribution function describes the relationship between observations in a sample space.
Probability distribution can be discrete or continuous. The discrete probability distribution deals with scenarios where the sample space is discrete (e.g. drawing a card from a deck, rolling a die...). On the opposite, if we are in scenarios where the sample space can take values in a continuos domain (e.g. measuring the height of adult humans) we are talking of continous probability distribution.
Instead of using the probability density function, we can describe the distribution using the cumulative distribution function (CDF), which calculates the probability of an observation equal or less than a value (i.e. \( F(x) = P(X \leq x) \) for some \(x\) ). In the case of a real-valued random variable, the cumulative distribution function has the following properties:
The following graph shows the simulation of two cumulative distributions functions. The blue line represent the discrete case, while the red line represent the continous one.