Definition:Random Variable/Discrete

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Definition

Definition 1

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\struct {S, \Sigma'}$ be a measurable space.

A discrete random variable on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S, \Sigma'}$ is a mapping $X: \Omega \to S$ such that:

$(1): \quad$ The image of $X$ is a countable subset of $S$
$(2): \quad$ $\forall x \in S: \set {\omega \in \Omega: \map X \omega = x} \in \Sigma$


Alternatively, the second condition can be written as:

$(2): \quad$ $\forall x \in S: X^{-1} \sqbrk {\set x} \in \Sigma$

where $X^{-1} \sqbrk {\set x}$ denotes the preimage of $\set x$.


Definition 2

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\struct {S, \Sigma'}$ be a measurable space.

Let $X$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S, \Sigma'}$.


Then we say that $X$ is a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S, \Sigma'}$ if and only if:

the image of $X$ is a countable subset of $\Omega'$.


Also known as

The image $\Img X$ of $X$ is often denoted $\Omega_X$.


Also see

  • Results about discrete random variables can be found here.