Math 2.0

definition

Random variable

Definition

Let (Ω,F,P)(\Omega, \mathcal{F}, P) be a probability space and let (E,E)(E, \mathcal{E}) be a set EE with a σ-algebra E\mathcal{E}. A random variable with values in EE is a function X ⁣:Ω→EX \colon \Omega \to E such that the preimage X−1(B)={ω∈Ω:X(ω)∈B}X^{-1}(B) = \{\omega \in \Omega : X(\omega) \in B\} is in F\mathcal{F} for each B∈EB \in \mathcal{E}. A real random variable is a random variable with values in R\mathbb{R}, with the Borel σ-algebra on R\mathbb{R}.

Remarks

The condition says that XX is a measurable function. It makes {X∈B}\{X \in B\} an event, so P(X∈B)P(X \in B) is defined. A real function XX on Ω\Omega is a random variable if and only if {X≤x}∈F\{X \le x\} \in \mathcal{F} for each x∈Rx \in \mathbb{R}. For two rolls of a die, Ω={1,…,6}2\Omega = \{1, \dots, 6\}^2 and the sum X(a,b)=a+bX(a, b) = a + b is a random variable. Sums, products and continuous functions of real random variables are random variables.

Explanations

L1 human

A random variable is a number that you read off from the result of the experiment. For two rolls of a die, the sum of the two numbers is a random variable. It is 77 for the result (3,4)(3, 4) and 1212 for the result (6,6)(6, 6).

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Sources

  1. Web https://en.wikipedia.org/wiki/Random_variable, Measure-theoretic definition

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