Probability theory MOC

Real random variable

A real random variable^[German Zufallsvariable] assigns a numerical value to an experimental outcome, that is world-state. In this way a random variable in the Probability model may be identified with a -measurable function¹

This turns out to be an incredibly useful concept, since it allows for a very natural comparison between outcomes. The notational convention is to use an uppercase letter for the random variable, in which case is used for specific values. Furthermore, itself is often used as a shorthand for where is the actual outcome (world-state). We can then define the probability of as follows

where a similar construction may be used for any other predicate. This definition naturally gives way to the distinction between a Discrete random variable and a Continuous random variable:

• The range of a Discrete random variable forms a set of cardinality
• If the probability of any exact value then is a Continuous random variable.
• Probability which are neither of these are mixed variables.
• One refers to values or ranges with nonzero probabilities as the support of

For both of these it is possible to define the following

• Independence of random variables
• Cumulative distribution function
• Expectation
• Standard deviation

Remarks

• A function of a random variable is a Random function
• Sum of independent random variables

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Footnotes

1. See also Multivariate random variable and the more General random variable ↩