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Discrete probability distribution

By definition a discrete probability distribution exists if the following two requirements are met.

  • It exists a set of random variables x_i \: i=\{1,2,2,...\} and each of these variables has a corresponding probability p_i with

p_i \ge 0 \; \text{for all} \; i=\{1,2,3..\}

  • All probabilities always sum up to 1

1 = \sum_i p_i

Special cases

Arbitrary finite number of random variables

Any arbitrary function that maps a probability to an arbitrary set of random variables is a probability distribution if the above mentioned requirements are met. For instance for the set of random variables x_1, x_2, x_3, x_4 the set of probabilities \{p_1=0.2,p_2=0.25,p_3=0.5,p_4=0.05\} are a discrete probability distribution as all these probabilities sum up to 1.

Two random variables

In this case, only one probability is needed as the other can be calculated by the sum which is always one.

p_1 = 1-p_2

This distribution is called Bernulli distribution.

One random variable

This is an exceptional case which is 100% certain. For instance the sun rises every morning. It might not be useful to consider such a case, as it has no practical consequences in statistics. It is however important to develop the fauceir abstraction of control.

Infinite number of random variables

The requirements are met if the sum of probabilities converges to 1 as in covergent series. Conclusively every series converging to 1 can be considered a random distribution whatever its practical implication.

Named discrete probability distributions

  • Bernulli distribution
  • Binomial distribution
  • Poisson distribution
  • Geometric distribution
  • Zipf distribution

Tags: Statistics

Categories: Mathematics


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