2.1 Probability spaces and random variables

Probability spaces form the foundation of statistical analysis, providing a framework to quantify uncertainty. They consist of sample spaces, events, and probability measures, allowing us to model real-world scenarios and make predictions based on available information.

Random variables bridge the gap between abstract probability spaces and concrete numerical values. By mapping outcomes to numbers, they enable mathematical modeling and statistical inference, making it possible to analyze complex probabilistic events and draw meaningful conclusions from data.

Fundamentals of Probability Spaces

Components of probability spaces

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• Sample space (Ω) encompasses all possible outcomes of an experiment forms the foundation for probability calculations (coin toss: heads, tails)
  • Elements are mutually exclusive and exhaustive ensuring comprehensive coverage of all possibilities
• Events represent subsets of the sample space allow for analysis of specific outcomes or combinations (rolling a die: even numbers)
  • Can be simple (single outcome) or compound (multiple outcomes) providing flexibility in probability analysis
• Probability measure (P) assigns numerical values to events quantifies likelihood of occurrences
  • Properties of probability measure ensure mathematical consistency:
    • Non-negativity: $P(A) \geq 0$ for any event A guarantees probabilities are always positive or zero
    • Normalization: $P(Ω) = 1$ establishes that the total probability of all possible outcomes is 1
    • Additivity: $P(A \cup B) = P(A) + P(B)$ for disjoint events A and B allows calculation of probabilities for combined events

Concept of random variables

• Random variable (X) maps outcomes from sample space to real numbers enables quantitative analysis of probabilistic events
  • Notation: $X: Ω \rightarrow \mathbb{R}$ formalizes the mapping process
• Purpose of random variables facilitates mathematical modeling and statistical inference
  • Quantify outcomes of experiments allowing for numerical representation (height of people)
  • Enable mathematical analysis of probabilistic events supporting complex calculations and predictions
• Types of random variables cater to different analytical needs:
  • Univariate: single variable focuses on one characteristic at a time (temperature)
  • Multivariate: multiple variables allows for analysis of relationships between variables (height and weight)

Types of Random Variables and Probability Calculations

Discrete vs continuous variables

• Discrete random variables take on countable number of distinct values suitable for scenarios with finite or countable outcomes
  • Examples: number of coin flips, dice rolls, binary outcomes (success/failure)
  • Probability mass function (PMF) used to describe probabilities assigns probability to each possible value
• Continuous random variables take on uncountable number of values in a range appropriate for measuring physical quantities
  • Examples: time, temperature, height
  • Probability density function (PDF) used to describe probabilities represents probability distribution over a continuous range
• Key differences highlight distinct mathematical approaches:
  • Summation vs integration for probability calculations reflects discrete vs continuous nature
  • PMF vs PDF for probability distributions determines method of probability assignment

Calculation using probability properties

• Complement rule: $P(A^c) = 1 - P(A)$ calculates probability of event not occurring
• Addition rule for mutually exclusive events: $P(A \cup B) = P(A) + P(B)$ applies when events cannot occur simultaneously
• General addition rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ accounts for overlap between events
• Conditional probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$ determines probability of A given B has occurred
• Multiplication rule: $P(A \cap B) = P(A|B) \cdot P(B)$ calculates joint probability of two events
• Law of total probability: $P(A) = \sum_{i} P(A|B_i) \cdot P(B_i)$ computes probability of A considering all possible scenarios
• Bayes' theorem: $P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$ updates probabilities based on new information