SAMPLE SPACE

Tags: #◇Computer Science #◇STT101A #◇Conceptual #◇Sample Space

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◇Overview

The Sample Space (denoted $S$ or $\Omega$) of a statistical experiment is the set of all possible elementary outcomes of that experiment. Every valid sample space must be both mutually exclusive (no two elementary outcomes can occur simultaneously on a single trial) and collectively exhaustive (every possible result of the experiment is represented by an element $\omega \in S$).

◇Key Definitions

• Sample Point (Elementary Outcome $\omega$): A single indivisible element $\omega \in S$.
• Finite Sample Space: Contains a finite number of outcomes $|S| = N < \infty$ (e.g., rolling two dice has $|S| = 36$).
• Countably Infinite (Discrete) Sample Space: Outcomes can be placed in one-to-one correspondence with $\mathbb{N} = \{1, 2, 3, \dots\}$ (e.g., tossing a coin until the first Head appears: $S = \{H, TH, TTH, TTTH, \dots\}$).
• Uncountable (Continuous) Sample Space: Outcomes fill a continuous interval or region of $\mathbb{R}^d$ (e.g., lifetime of a solid-state drive: $S = [0, \infty)$).

◇Worked Example

Problem: Specify the sample space $S$ and its type for three experiments:

1. Experiment 1: Tossing a coin and rolling a 4-sided die:
   $S_1 = \{(H,1), (H,2), (H,3), (H,4), (T,1), (T,2), (T,3), (T,4)\} \quad (\text{Finite discrete, } |S_1| = 8)$
2. Experiment 2: Counting the number of packet retransmissions until a frame is acknowledged:
   $S_2 = \{0, 1, 2, 3, \dots\} \quad (\text{Countably infinite discrete})$
3. Experiment 3: Measuring the voltage across a resistor between $0\text{ V}$ and $5\text{ V}$:
   $S_3 = [0, 5] = \{v \in \mathbb{R} : 0 \le v \le 5\} \quad (\text{Uncountable continuous})$

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