An example of the distribution function for a discrete random variable is shown in Figure 3.8. Observe that there are two distinguishing features of this type of distribution function:

FIGURE 3.8 Distribution function for a discrete (Bernoulli) random variable.

  • FX(x) changes instantaneously at values of x that are outcomes of the random variable.
  • FX(x) remains constant between these outcomes.

There are two useful ways of representing the outcomes and the probability assignment for a discrete random variable, using either the Kronecker delta function or the Dirac delta function . The distribution function in Figure 3.8 corresponds to the symmetric Bernoulli random variable with equally likely outcomes . This random variable would arise, for example, when tossing a single fair coin with the mapping: and .

The Kronecker delta function is convenient for representing the outcomes of a discrete random variable because there are no continuity issues. Moreover, ...

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