Editing Probability density function (section)

==Link between discrete and continuous distributions==

It is possible to represent certain discrete random variables as well as random variables involving both a continuous and a discrete part with a [[Generalized function|generalized]] probability density function using the [[Dirac delta function]]. (This is not possible with a probability density function in the sense defined above, it may be done with a [[Distribution (mathematics)|distribution]].) For example, consider a binary discrete [[random variable]] having the [[Rademacher distribution]]—that is, taking −1 or 1 for values, with probability {{1/2}} each. The density of probability associated with this variable is:
<math display="block">f(t) = \frac{1}{2} (\delta(t+1)+\delta(t-1)).</math>

More generally, if a discrete variable can take {{mvar|n}} different values among real numbers, then the associated probability density function is:
<math display="block">f(t) = \sum_{i=1}^n p_i\, \delta(t-x_i),</math>
where <math>x_1, \ldots, x_n</math> are the discrete values accessible to the variable and <math>p_1, \ldots, p_n</math> are the probabilities associated with these values.

This substantially unifies the treatment of discrete and continuous probability distributions. The above expression allows for determining statistical characteristics of such a discrete variable (such as the [[mean]], [[variance]], and [[kurtosis]]), starting from the formulas given for a continuous distribution of the probability.