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 𝑋 in the Probability model (𝜉,F,ℙ) may be identified with a ℙ-measurable function¹

𝑋:𝜉→ℝ

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 𝑋 for the random variable, in which case 𝑥 is used for specific values. Furthermore, 𝑋 itself is often used as a shorthand for 𝑋(𝑠) where 𝑠 ∈𝜉 is the actual outcome (world-state). We can then define the probability of 𝑋 =𝑥 as follows

ℙ(𝑋=𝑥)=ℙ(𝑋(𝑠)=𝑥)=ℙ({𝑠∈𝜉∣𝑋(𝑠)=𝑠})

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 ≤ℵ0
• If the probability of any exact value ℙ(𝑋 =𝑥) =0 then 𝑋 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 𝑋

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 ↩