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Probability Measures ›

Topological Sigma Algebra ›

Measurable Functions ›

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Central Limit Theorem ›

Conditional Dependency Graph ›

Cumulative Distribution Functions ›

Distortion Functions ›

Estimates ›

Independent Random Variables ›

Random Real Vectors ›

Random Variable Laws ›

Random Variable Sigma Algebras ›

Real-Valued Random Variable Expectation ›

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Random Variables

Definition

A random variable is a measurable map from a probability space to a measurable space.

A real-valued random variable is a measurable map between the probability space and the set of real numbers with its topological sigma algebra. We so frequently work with real-valued random variables that if the range the random variable is not specified, it is assumed to be a subset of $\R $.

Notation

Suppose $(X, \mathcal{A} , \mathbfsf{P} )$ is a probability space and $(Y, \mathcal{B} )$ is a measurable space. A random variable is a measurable function $f: X \to Y$.

Some authors tend to denote real-valued random variables by upper case Latin letters: for example, $X, Y, Z$. They reserve lower case letters $x$, $y$, $z$ for the elements of the codomains of these functions. In such cases, they often denote the set of outcomes as $\Omega $, which we have mentioned is a mnemonic for outcomes.

Special notation for common cases

Some authors use notation for the probability of particular, common sets. Suppose $(\Omega , \mathcal{A} , \mathbfsf{P} )$ is a probability space with $X: \Omega \to \R $ a real-valued random variable. Many authors use $\mathbfsf{P} (X \in A)$ (or $P(X \in A)$, no bold) to denote

\[ \mathbfsf{P} (X^{-1}(A)) = \mathbfsf{P} (\Set*{\omega \in \Omega }{X(\omega ) \in A}) \]

where $A \in \mathcal{B} (\R )$, the Borel sigma algebra on $\R $. We will tend to use brackets in place of parentheses for clarity. So we will write $\mathbfsf{P} [ X \in A ]$.

Similar to the above, suppose $Y: \Omega \to \R $ is a random variable and $B \in \mathcal{B} (\R )$. Then we will use $\mathbfsf{P} [X \in A, Y \in B]$ to denote

\[ \mathbfsf{P} (X^{-1}(A) \cap Y^{-1}(B)) = \mathbfsf{P} (\Set*{\omega \in \Omega }{X(\omega ) \in A \text{ and } Y(\omega ) \in B}). \]

Similarly for $n$ random variables $X_1, \dots , X_n: \Omega \to \R $, and Borel sets $A_1, \dots , A_n$, we will use $\mathbfsf{P} [X_1 \in A_1, \dots , X_n \in A_n]$ to denote $\mathbfsf{P} (\cap _{i = 1}^{n} X_i^{-1}(A_i))$.