Actuarium

SOA PProbability

Associateship (ASA)
3 hoursยท30 multiple-choiceยทโ‰ˆ300 study hours

Syllabus learning objectives

Official syllabus

Paraphrased from the SOA syllabus so the study-plan builder and practice sets track the topics you will actually be examined on. Weights are the official topic ranges.

General probability & random variables

14%
  • Apply set theory, counting techniques and axioms of probability to compute event probabilities.
  • Compute conditional probabilities, apply Bayes' theorem and assess independence of events.
  • Derive distributions, expectations and variances of functions of random variables, including transformations.
  • Compute moment generating functions and use them to find moments and identify distributions.

Univariate distributions

43%
  • Compute probabilities, moments, percentiles and MGFs for binomial, negative binomial, geometric, hypergeometric and Poisson distributions.
  • Compute probabilities, moments, percentiles and MGFs for uniform, exponential, gamma, and normal distributions.
  • Apply the memoryless property of the exponential/geometric distributions and relationships between distributions (e.g., Poisson-gamma mixture).
  • Work with mixed distributions and distributions with a probability mass at a point.

Multivariate distributions

43%
  • Compute joint, marginal and conditional distributions for discrete and continuous random variables.
  • Compute covariance and correlation, and determine independence of jointly distributed random variables.
  • Derive the distribution of a sum of independent random variables via convolution or MGFs.
  • Apply the double-expectation and conditional-variance formulas to compute moments of mixtures and compound variables.

Lecture videos for this exam

Open the full video library โ†’

Exam P Crash Course - Part 1/6 (General Probability)

Jeff Yang, FSA ยท Probability (Exam P)

Basic Probability Part 1 (SOA Exam P โ€“ Probability โ€“ General Probability Module )

AnalystPrep ยท Probability (Exam P)

Law of Total Probability โ€‹(SOA Exam P โ€“ Probability โ€“ General Probability Module)

AnalystPrep ยท Probability (Exam P)

Statistics 110: Probability

Harvard University ยท Probability (Exam P) ยท playlist

6.041 Probabilistic Systems Analysis and Applied Probability

MIT OpenCourseWare ยท Probability (Exam P) ยท playlist

Overview

Exam P/1 tests calculus-based probability: set theory, conditional probability and Bayes, discrete and continuous univariate distributions, transformations, moment generating functions, and joint/conditional/marginal distributions. It is shared with the SOA and is the usual first exam.

Duration
3 hours
Questions
30 multiple-choice
Style
Computer-based testing, 5 answer choices
Passing
Scaled score 6 of 10

Syllabus map

General probability & random variables
14%
Probability and Distributions for Insurance
Univariate distributions
43%
Probability and Distributions for Insurance
Multivariate distributions
43%
Probability and Distributions for InsuranceDependence and Copulas

Key formulas

Bayes' theorem P(AiโˆฃB)=P(BโˆฃAi)P(Ai)โˆ‘jP(BโˆฃAj)P(Aj)P(A_i\mid B)=\frac{P(B\mid A_i)P(A_i)}{\sum_j P(B\mid A_j)P(A_j)}

Conditional variance (law of total variance) Varโก(X)=E[Varโก(XโˆฃY)]+Varโก(E[XโˆฃY])\operatorname{Var}(X)=E[\operatorname{Var}(X\mid Y)]+\operatorname{Var}(E[X\mid Y])

MGF moments โ€…โ€ŠE[Xk]=MX(k)(0)\;E[X^k]=M_X^{(k)}(0); sum of independent variables: MX+Y(t)=MX(t)MY(t)M_{X+Y}(t)=M_X(t)M_Y(t).

Expected payment with deductible dd on loss XX (per loss): E[(Xโˆ’d)+]=โˆซdโˆžSX(x)โ€‰dxE[(X-d)_+]=\int_d^\infty S_X(x)\,dx.

Exponential memorylessness โ€…โ€ŠP(X>s+tโˆฃX>s)=P(X>t)\;P(X>s+t\mid X>s)=P(X>t); for XโˆผExp(ฮธ)X\sim\text{Exp}(\theta), E[(Xโˆ’d)+]=ฮธeโˆ’d/ฮธE[(X-d)_+]=\theta e^{-d/\theta}.

Order statistics for i.i.d. sample of size nn: F(n)(x)=F(x)nF_{(n)}(x)=F(x)^n, F(1)(x)=1โˆ’(1โˆ’F(x))nF_{(1)}(x)=1-(1-F(x))^n.

Study strategy

  1. Memorize the distribution table (pmf/pdf, mean, variance, MGF) for binomial, Poisson, geometric, negative binomial, uniform, exponential, gamma, normal, and lognormal โ€” the exam does not provide one.

  2. Drill conditional probability and Bayes with insurance framing (risk classes, claim/no-claim) until the tree diagram is automatic.

  3. Practice double integrals over triangular regions; most multivariate misses are limits-of-integration errors.

  4. Time budget: 6 minutes per question; flag and move on after 4.

Common traps

  • Confusing per-loss and per-payment expectations under a deductible.

  • Using ฯƒ2\sigma^2 where the question gives ฯƒ\sigma (and vice versa) for the normal.

  • Forgetting the Jacobian when transforming a continuous variable.

  • Variance of a sum: dropping the covariance term for dependent variables.

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