[Exam P Syllabus - November 2025](https://www.soa.org/49facb/globalassets/assets/files/edu/2025/fall/syllabi/2025-11-exam-p-syllabus.pdf) | Domain | Domain Weight | | -------------------------------- | ------------- | | 1. General Probability | 23-30% | | 2. Univariate Random Variables | 44-50% | | 3. Multivariate Random Variables | 23-30% | **Prerequisite Knowledge** 1. Calculus, including series, differentiation, and integration. 2. Concepts introduced in “[Risk and Insurance](https://www.soa.org/49355c/globalassets/assets/files/edu/p-21-05.pdf).” ### 1. General Probability > **Probability** is the mathematical framework that quantifies uncertainty by assigning numerical values (between 0 and 1) to the likelihood of events occurring. | Term | Definition | | ------------------------------------ | ------------------------------------------------------------------------------------------------ | | [[Set Function]] | A function assigning values to sets. | | [[Venn Diagram]] | Diagram showing relationships between sets using overlapping circles. | | [[Sample Space]] | The set of all possible outcomes. | | [[Event]] | A subset of the sample space. | | [[Probability]] | A set function on a collection of events. | | [[Axioms of Probability]] | Fundamental rules defining probability, including non-negativity, normalization, and additivity. | | [[Probability Addition Rules]] | Rules for summing probabilities of events. | | [[Probability Multiplication Rules]] | Rules for multiplying probabilities of dependent or independent events. | | [[Independent Events]] | Events whose outcomes don’t affect each other. | | [[Mutually Exclusive Events]] | Events that cannot occur simultaneously. | | [[Conditional Probability]] | Probability of an event given another has occurred. | | [[Combinatorics]] | Study of counting, arrangements, and combinations. | | [[Combination]] | Selection of items where order doesn’t matter. | | [[Permutation]] | Arrangement of items where order matters. | | [[Bayes Theorem]] | Updates probabilities based on new evidence. | | [[The Law of Total Probability]] | Computes event probability by considering all possible scenarios. | ### 2. Univariate Random Variables > A **Univariate Random Variable** is a single random variable that represents uncertain outcomes of an experiment, taking values in one dimension. | Term | Definition | | ------------------------------------------ | ------------------------------------------------------------------------------------------------ | | [[Random Variable]] | A function mapping outcomes to numerical values. | | [[Probability Density Function (PDF)]] | A function defining the probability distribution of a continuous random variable. | | [[Cumulative Distribution Function (CDF)]] | A function giving the probability that a random variable is less than or equal to a given value. | | [[Conditional Probability]] | The probability of an event given that another event has occurred. | | [[Expected Value]] | The weighted average of all possible values of a random variable. | | [[Moments]] | Quantities describing the shape of a distribution, including mean and variance. | | [[Mode]] | The value(s) with the highest probability or frequency. | | [[Median]] | The middle value that splits the probability distribution in half. | | [[Percentile]] | A value below which a given percentage of observations fall. | | [[Variance]] | The expected value of the squared deviation from the mean. | | [[Standard Deviation (SD)]] | The square root of the variance, measuring dispersion. | | [[Coefficient of Variation]] | The ratio of the standard deviation to the mean, indicating relative variability. | | [[Policy Adjustments]] | Changes to policy terms affecting coverage, limits, or conditions. | | [[Deductible]] | The amount the insured pays before insurance coverage begins. | | [[Coinsurance]] | A percentage of the covered loss shared by the insured after the deductible. | | [[Benefit Limits]] | The maximum amount payable under an insurance policy. | | [[Payment]] | The amount paid by the insurer for a covered claim. | | [[Inflation]] | The increase in the price level over time, affecting claim costs. | | [[Loss Random Variable]] | A random variable representing the financial loss from an event. | | [[Payment Random Variable]] | A random variable representing the amount paid by the insurer. | ### 3. Multivariate Random Variables > A **Multivariate Random Variable** is a collection of two or more random variables considered jointly, capturing the relationships and dependencies among them. | Term | Definition | | ---------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------- | | [[Joint Probability Function]] (discrete) | A function giving the probability of simultaneous outcomes of two or more discrete random variables. | | [[Join Cumulative Distribution Function]] (discrete) | A function giving the probability that two or more random variables are less than or equal to specific values. | | [[Conditional Probability Function]] (discrete) | A function giving the probability of one discrete variable given another. | | [[Marginal Probability Function]] (discrete) | A function giving the probabilities of individual variables by summing over others. | | [[Conditional Probability]] | The probability of an event occurring given that another event has already occurred. | | Joint Moments (discrete) | The expected values of products of powers of two or more discrete random variables. | | [[Covariance]] | A measure of how two random variables change together. | | [[Correlation coefficient]] | A normalized measure of the strength and direction of a linear relationship between two variables. | | [[Order Statistics (independent set)]] | Statistics derived from the ordered values of a sample. | | [[Linear Combination]] (independent set, normal) | A weighted sum of independent normal random variables, which is itself normally distributed. | | [[Moments]] | Measures describing the shape of a distribution, including central and non-central moments. | | [[Central Limit Theorem (CLT)]] | A theorem stating that the sum (or average) of a large number of independent random variables approaches a normal distribution. | | [[Linear Combinations (IID)]] | A weighted sum of identically distributed, independent random variables. | ## Sources | Source | Coverage | | ----------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------- | | [[A First Course in Probability (Ross - 2019)]] | Chapters 1-8^[Excluding 4.8.4, 5.6.2, 5.6.3, 5.6.5, 5.7, 7.2.1, 7.2.2, 7.3, 7.6, 7.7, 7.8, 7.9]<br> | | [[Mathematical Statistics with Applications - 2008]] | Chapters 1-8^[Exlcuding 2.12, MGF, 4.10, Continuous Multivariate Distributions, 5.10, 7.4] | | [[Probability for Risk Management (Hassett - 2021)]] | Chapters 1-11 | | [[Probability and Statistics with Applications-- A Problem-Solving Text (Asimow - 2021)]] | Chapters 1-8 | | [[Probability and Statistical Inference (Hogg - 2020)]] | Chapters 1-5 | | [[Probability (Leemis - 2018)]] | Chapters 1-8 |