Math 2.0

definition

Probability distribution

Definition

Let EE be a set with a σ-algebra E\mathcal{E}. A probability distribution on (E,E)(E, \mathcal{E}) is a probability measure μ\mu on (E,E)(E, \mathcal{E}). A probability distribution on R\mathbb{R} is a probability measure on R\mathbb{R} with its Borel σ-algebra B(R)\mathcal{B}(\mathbb{R}).

Remarks

Each random variable XX with values in EE gives a distribution, its law B↦P(X∈B)B \mapsto P(X \in B). Conversely, each distribution μ\mu is the law of some random variable, for example the identity map on (E,E,μ)(E, \mathcal{E}, \mu). A distribution on R\mathbb{R} is determined by its cumulative distribution function. Important types are discrete distributions, given by a probability mass function, and absolutely continuous distributions, given by a probability density function.

Explanations

L1 human

A distribution tells you which values are possible and how likely each range of values is. For a fair die, the distribution gives each of the values 1,…,61, \dots, 6 the probability 1/61/6.

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  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

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Sources

  1. Web https://en.wikipedia.org/wiki/Probability_distribution, General probability definition

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