Probability distribution: Difference between revisions

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==Introduction==

A probability distribution is a mathematical description of the probabilities of events, subsets of the [[sample space]]. The sample space, often represented in notation by <math>\ \Omega\ ,</math> is the [[Set (mathematics)|set]] of all possible [[outcome (probability)|outcomes]] of a random phenomenon being observed; it may be any set: Aa set of [[real numbers]], a set of descriptive labels, a set of [[vector (mathematics)|vectors]], a set of arbitrary non-numerical values, etc. For example, the sample space of a coin flip could be {{math|{{nobr|&ensp;Ω {{=}} {{big|<nowiki>{</nowiki>}} "heads", "tails" {{big|<nowiki>}</nowiki>}} }}.}}

To define probability distributions for the specific case of [[random variables]] (so the sample space can be seen as a numeric set), it is common to distinguish between '''discrete''' and '''absolutely continuous''' [[random variable]]s. In the discrete case, it is sufficient to specify a [[probability mass function]] <math>\ p\ </math> assigning a probability to each possible outcome: for example, when throwing a fair [[dice]], each of the six digits {{math|“1”}} to {{math|“6”}}, corresponding to the number of dots on the die, has the probability <math>\ \tfrac{1}{6} ~.</math> The probability of an [[Event (probability theory)|event]] is then defined to be the sum of the probabilities of all outcomes that satisfy the event; for example, the probability of the event "the die rolls an even value" is