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 probability density function, by using the Dirac delta function. For example, let us consider a binary discrete random variable taking −1 or 1 for values, with probability ½ each.
The density of probability associated with this variable is:
More generally, if a discrete variable can take n different values among real numbers, then the associated probability density function is:
where x1, …, xn are the discrete values accessible to the variable and p1, …, pn are the probabilities associated with these values.
This substantially unifies the treatment of discrete and continuous probability distributions. For instance, the above expression allows for determining statistical characteristics of such a discrete variable (such as its mean, its variance and its kurtosis), starting from the formulas given for a continuous distribution of the probability.
Read more about this topic: Probability Density Function
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