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4.18. Probability Theory (PRO)

This knowledge area covers the mathematical foundations of probability, including sample spaces, random variables, probability distributions, joint distributions, moments, transformations, and the fundamental limit theorems. It provides the theoretical basis for all of statistical inference and modeling.

Knowledge Area (KA)	Core Tier1	Core Tier2
4.18.1 Sample Spaces and Events	1	1
4.18.2 Probability Axioms and Properties	1	1
4.18.3 Conditional Probability and Independence	1	1
4.18.4 Discrete Random Variables and Distributions	1	1
4.18.5 Continuous Random Variables and Distributions	1	1
4.18.6 Joint, Marginal, and Conditional Distributions	1	1
4.18.7 Limit Theorems: LLN and CLT	1	1

4.18.1. PRO/Sample Spaces and Events  (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top

Formal definition of random experiments, sample spaces, and events. Covers set-theoretic operations on events and the construction of sigma-algebras as the foundation of probability theory.
Topics:
Core

• Random experiments, outcomes, and sample spaces: finite, countable, and uncountable
• Set-theoretic operations on events: union, intersection, complement, and De Morgan's laws
• Sigma-algebras and measurable spaces as the foundation for rigorous probability
• Borel sets and the Borel sigma-algebra on the real line

Learning Outcomes:
Core:

1. Define sample spaces, events, and sigma-algebras for a given random experiment [Familiarity]
2. Apply set-theoretic operations to compute probabilities of compound events [Usage]
3. Construct appropriate sigma-algebras for measurable spaces arising in statistical problems [Assessment]

4.18.2. PRO/Probability Axioms and Properties  (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top

Kolmogorov's axioms of probability and derived properties. Covers classical, frequentist, and subjective interpretations of probability and their implications.
Topics:
Core

• Kolmogorov's three axioms of probability and their consequences
• Derived properties: complement rule, addition rule, monotonicity, and continuity of probability
• Classical, frequentist, and subjective (Bayesian) interpretations of probability
• Counting methods: permutations, combinations, and the multiplication principle

Learning Outcomes:
Core:

1. State Kolmogorov's axioms and derive fundamental probability properties [Familiarity]
2. Calculate probabilities of events using the addition rule and complement rule [Usage]
3. Apply counting methods to compute probabilities in finite sample spaces [Usage]

4.18.3. PRO/Conditional Probability and Independence  (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top

Conditional probability, the multiplication rule, the law of total probability, Bayes' theorem, and the concept of independence between events and collections of events.
Topics:
Core

• Definition and interpretation of conditional probability P(A|B) and its properties
• Bayes' theorem and the law of total probability: derivation and applications
• Independence of events: pairwise and mutual independence
• Conditional independence and its role in probabilistic graphical models

Learning Outcomes:
Core:

1. Explain conditional probability and independence and their relationship [Familiarity]
2. Apply Bayes' theorem and the law of total probability to solve probability problems [Usage]
3. Analyze whether events are independent or conditionally independent in a given probability model [Assessment]

4.18.4. PRO/Discrete Random Variables and Distributions  (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top

Random variables as measurable functions, probability mass functions, cumulative distribution functions, and the most important discrete probability distributions.
Topics:
Core

• Probability mass functions (PMF) and cumulative distribution functions (CDF) for discrete random variables
• Key discrete distributions: Bernoulli, Binomial, Geometric, Negative Binomial, Poisson, Hypergeometric
• Properties of discrete distributions: mean, variance, skewness, and kurtosis
• Poisson approximation to the Binomial and limiting distribution arguments

Learning Outcomes:
Core:

1. Identify and describe the key discrete probability distributions and their parameters [Familiarity]
2. Calculate probabilities, means, and variances using PMFs of standard discrete distributions [Usage]
3. Select an appropriate discrete distribution to model a given random phenomenon and justify the choice [Assessment]

4.18.5. PRO/Continuous Random Variables and Distributions  (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top

Probability density functions, CDFs, and the most important continuous probability distributions used in statistical theory and practice.
Topics:
Core

• Probability density functions (PDF) and CDFs for continuous random variables
• Key continuous distributions: Uniform, Normal, Exponential, Gamma, Beta, Chi-squared, t, F
• The Normal distribution: properties, standard normal, and its central role in statistical inference
• The exponential family of distributions: canonical form and sufficient statistics

Learning Outcomes:
Core:

1. Describe the properties and parameters of key continuous probability distributions [Familiarity]
2. Calculate probabilities and quantiles for standard continuous distributions [Usage]
3. Identify distributions belonging to the exponential family and derive their sufficient statistics [Assessment]

4.18.6. PRO/Joint, Marginal, and Conditional Distributions  (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top

Joint distributions of multiple random variables, marginal and conditional distributions, covariance, correlation, and the bivariate normal distribution.
Topics:
Core

• Joint PMFs and PDFs for two or more random variables
• Marginal and conditional distributions derived from joint distributions
• Covariance, correlation, and their properties; independence implies zero covariance (converse not always true)
• The bivariate normal distribution and its conditional distributions

Learning Outcomes:
Core:

1. Derive marginal and conditional distributions from a given joint distribution [Usage]
2. Calculate covariance and correlation between two random variables [Usage]
3. Analyze the conditional structure of the bivariate normal distribution [Assessment]

4.18.7. PRO/Limit Theorems: LLN and CLT  (Core Tier1: 1 hr, Core Tier2: 1 hr) ↑ Back to top

The laws of large numbers and the central limit theorem as the mathematical justification for statistical estimation and inference procedures based on large samples.
Topics:
Core

• Weak law of large numbers: convergence in probability and its implications for estimation
• Central limit theorem: statement, conditions, and role in justifying normal approximations
• Applications of the CLT: normal approximation to the binomial and sampling distributions
• Strong law of large numbers: almost sure convergence
• The delta method: asymptotic distribution of smooth functions of estimators

Learning Outcomes:
Core:

1. State the LLN and CLT and explain the conditions under which they hold [Familiarity]
2. Apply the CLT to derive normal approximations and sampling distributions [Usage]
3. Apply the delta method to derive asymptotic distributions of nonlinear estimators [Assessment]