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DISCRETE PROBABILITY
It can be defined as the probability distribution characterized by a Probability mass function where the distribution of a random variable X is discrete, then X is known as discrete random variable, if
Where u runs through the set of all possible values of X and it follows that such a random variable can assume the only a finite set of values. This set of possible values is a topologically discrete set in the sense that all its points are the isolated point, but there is discrete random variable for which this countable set is dense set on the real line.
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Among the most well-known distributions of this type which are used for statistical modeling are the Poisson distribution and the Bernoulli distribution as well as the Geometric distribution and the Negative binomial distribution. In addition, the uniform distribution is commonly used in computer programs which are used to make equal-probability random selections between a number of choices.
Cumulative density is a discrete random variable which can be defined as a random variable whose cumulative distribution function that is cdf increases only by jump discontinuity. This means that its cdf increases only where it "jumps" to a higher value and it is a constant between those jumps.
The points where it jumps are precisely the values which the random variable may take and the number of such jumps may be finite as well as countably infinite. The set of locations representing such jumps need not be topologically discrete and for example, the cdf might jump at each rational number.
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Delta-function representation is often represented as a generalized probability density function which involves the Dirac delta function which substantially unifies the treatment of continuous as well as, discrete distributions. This is useful when dealing with probability distributions involving both a continuous as well as a discrete part.
Indicator-function representation
For a discrete random variable X, let u0,u1, ... be the values with non-zero probability and it is denoted as follows
These are disjoint set and represented by the formula as below
And it follows that the probability that X takes any value except for u0,u1, ... is zero, and thus X can be written as shown below
Except on a set of probability zero, 1A is the indicator function of A which serves as an alternative definition of discrete random variables.
