Probability Theory Fundamentals

Core probability concepts including axioms, conditional probability, Bayes theorem, and random variables.

Key Concepts

• •Axioms of probability: sample spaces, events, and the three axioms (non-negativity, normalization, additivity)
• •Conditional probability: P(A|B) = P(A and B) / P(B), with applications to insurance claims
• •Bayes theorem: updating probabilities with new information, widely used in credibility theory
• •Random variables: discrete and continuous, with PMFs, PDFs, and CDFs
• •Expected value and variance: E[X], Var(X), and their properties under linear transformations
• •Common distributions: binomial, Poisson, normal, exponential, gamma, beta, uniform
• •Joint distributions: marginal and conditional distributions, independence
• •Moment generating functions: M(t) = E[e^(tX)], used to identify distributions and find moments
• •Transformations: CDF technique, Jacobian method for functions of random variables
• •Law of large numbers and central limit theorem: convergence results for sums of random variables

Study Tips

• 1.Start with combinatorics and counting principles before moving to probability axioms.
• 2.Practice computing conditional probabilities using tree diagrams and Bayes theorem.
• 3.Memorize the key properties of each distribution: mean, variance, MGF, and special relationships.
• 4.Work through joint distribution problems methodically: set up the region of integration carefully.
• 5.Use moment generating functions to verify distribution identities and find moments efficiently.

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