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Random Variable

Formula

\[ X:\Omega \to \mathbb{R} \]

Parameters

  • \(\Omega\): sample space (all outcomes)
  • \(X\): function assigning a numeric value to each outcome

What it means

A random variable converts outcomes of a random experiment into numbers so we can compute probabilities, averages, and other statistics.

What it's used for

  • Defining distributions, expectations, and variances.
  • Modeling numeric uncertainty from experiments or data.

Key properties

  • Can be discrete, continuous, or mixed.
  • Probability statements like \(P(X \le x)\) are defined from the underlying experiment.

Common gotchas

  • A random variable is a function, not "the outcome itself."
  • Different random variables can be defined on the same sample space.

Example

If \(\Omega=\{\text{HH},\text{HT},\text{TH},\text{TT}\}\) for two coin flips, let \(X\) be the number of heads. Then \(X(\text{HT})=1\), \(X(\text{HH})=2\).

How to Compute (Pseudocode)

Input: sample space outcomes and a numeric mapping rule X
Output: random-variable values X(omega)

for each outcome omega in the sample space representation:
  assign X(omega) according to the modeling rule
return the random variable mapping (or sampled values under that mapping)

Complexity

  • Time: Depends on how outcomes are represented and whether you are defining the mapping symbolically or evaluating it on sampled outcomes
  • Space: Depends on whether the mapping is symbolic or materialized over many outcomes/samples
  • Assumptions: Conceptual definition card; practical computation usually happens via sampled data or a known distribution model

See also