RANDOM VARIABLE

Tags: #◇Computer Science #◇STT101A #◇Conceptual #◇Random Variable

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◇Overview

A Random Variable $X$ is neither random nor a variable in the algebraic sense: formally, it is a deterministic real-valued function $X : S \to \mathbb{R}$ defined on a sample space $S$ that maps every elementary outcome $\omega \in S$ to a unique real number $X(\omega) \in \mathbb{R}$. Randomness enters through the underlying probability measure $P$ on $S$.

◇Key Definitions

• Function Mapping: $X : S \to \mathbb{R}$, assigning a numerical value $X(\omega)$ to each outcome $\omega \in S$.
• Inverse Image (Pre-image) Probability: For any subset $B \subseteq \mathbb{R}$, the induced probability is:
  $P(X \in B) = P(\{\omega \in S : X(\omega) \in B\}) = P(X^{-1}(B))$
• Notation Convention: Uppercase letters ($X, Y, Z$) denote the random variable (the function), while lowercase letters ($x, y, z$) denote specific numerical values in its range/support.

◇Worked Example

Problem: Flip a fair coin $3$ times, so $S = \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$ ($|S| = 8$). Let $X(\omega)$ be the number of Heads in outcome $\omega$.

• $X(TTT) = 0 \implies P(X = 0) = \frac{1}{8}$
• $X(HTT) = X(THT) = X(TTH) = 1 \implies P(X = 1) = \frac{3}{8}$
• $X(HHT) = X(HTH) = X(THH) = 2 \implies P(X = 2) = \frac{3}{8}$
• $X(HHH) = 3 \implies P(X = 3) = \frac{1}{8}$

◇References & Navigation

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• Related Notes: ◇Discrete Probability Distribution, ◇Bayes's Rules