SOA Exam P Cheat Sheet

SOA Exam P Cheat Sheet #

Are you preparing for the Society of Actuaries (SOA) Exam P? This comprehensive cheat sheet covers all the essential concepts you need to master for probability and statistics. Bookmark this page and use it as your quick reference guide while studying!

Table of Contents #

  1. Key Information About Exam P
  2. Essential Probability Concepts
  3. Random Variables
  4. Advanced Concepts
  5. Transformation Techniques
  6. Study Tips
  7. Exam Day Preparation

Key Information About Exam P #

The SOA Exam P (Probability) is the foundational exam for aspiring actuaries, testing comprehensive knowledge of probability theory and its applications. Understanding this exam’s structure is crucial for effective preparation.

Exam Structure:

  • Duration: 3 hours
  • Number of questions: 30 multiple-choice questions
  • Passing score: Typically around 70% (varies by sitting)
  • Calculator: Only BA-II Plus or BA-II Plus Professional allowed
  • Format: Computer-based testing (CBT)
  • Cost: $225 USD (as of 2024)

Topics Covered: The exam covers probability theory fundamentals including general probability concepts, univariate probability distributions, multivariate probability distributions, and coverage modifications. The syllabus emphasizes both theoretical understanding and practical application of probability concepts that actuaries use in their daily work.

Essential Probability Concepts #

Basic Probability Rules #

Mastering fundamental probability rules is essential for success on Exam P. These rules form the foundation for all advanced probability calculations.

Addition Rule (Union of Events): For any two events A and B:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

For mutually exclusive events (A ∩ B = ∅):

P(A ∪ B) = P(A) + P(B)

Multiplication Rule (Intersection of Events): For dependent events:

P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B)

For independent events:

P(A ∩ B) = P(A) × P(B)

Complement Rule:

P(A^c) = 1 - P(A)

De Morgan’s Laws:

(A ∪ B)^c = A^c ∩ B^c
(A ∩ B)^c = A^c ∪ B^c

Conditional Probability #

Conditional probability represents the likelihood of an event occurring given that another event has already occurred. This concept is fundamental to actuarial applications.

Basic Formula:

P(A|B) = P(A ∩ B) / P(B), where P(B) > 0

Law of Total Probability: If {B₁, B₂, …, Bₙ} is a partition of the sample space, then:

P(A) = Σ P(A|Bᵢ) × P(Bᵢ)

Independence: Events A and B are independent if and only if:

P(A|B) = P(A) or equivalently P(A ∩ B) = P(A) × P(B)

Bayes’ Theorem #

Bayes’ Theorem allows us to update probabilities based on new information, making it incredibly valuable for actuarial applications involving risk assessment and claims analysis.

Basic Formula:

P(A|B) = [P(B|A) × P(A)] / P(B)

Extended Form with Partition: If {A₁, A₂, …, Aₙ} is a partition of the sample space, then:

P(Aᵢ|B) = [P(B|Aᵢ) × P(Aᵢ)] / [Σ P(B|Aⱼ) × P(Aⱼ)]

Practical Application: Bayes’ Theorem is frequently used in insurance for updating claim probabilities based on policyholder characteristics or loss experience.

Random Variables #

Types of Random Variables #

Understanding the distinction between discrete and continuous random variables is crucial for selecting appropriate probability models and calculation methods.

Discrete Random Variables:

  • Take on countable values (finite or countably infinite)
  • Probability Mass Function (PMF): P(X = x)
  • Properties of PMF:
    • P(X = x) ≥ 0 for all x
    • Σ P(X = x) = 1
  • Cumulative Distribution Function (CDF): F(x) = P(X ≤ x) = Σ P(X = k) for k ≤ x

Continuous Random Variables:

  • Take on any value in an interval (uncountably infinite)
  • Probability Density Function (PDF): f(x)
  • Properties of PDF:
    • f(x) ≥ 0 for all x
    • ∫₋∞^∞ f(x)dx = 1
  • CDF: F(x) = P(X ≤ x) = ∫₋∞^x f(t)dt
  • Important: P(X = x) = 0 for any specific value x

Expected Value and Variance #

These measures provide essential information about the central tendency and variability of random variables, which are fundamental to actuarial modeling.

Expected Value (Mean):

  • Discrete: E(X) = μ = Σ x × P(X = x)
  • Continuous: E(X) = μ = ∫₋∞^∞ x × f(x)dx

Properties of Expected Value:

  • E(aX + b) = aE(X) + b
  • E(X + Y) = E(X) + E(Y) (always true)
  • E(XY) = E(X)E(Y) (only if X and Y are independent)

Variance:

Var(X) = σ² = E[(X - μ)²] = E(X²) - [E(X)]²

Properties of Variance:

  • Var(aX + b) = a²Var(X)
  • Var(X + Y) = Var(X) + Var(Y) + 2Cov(X,Y)
  • If X and Y are independent: Var(X + Y) = Var(X) + Var(Y)

Standard Deviation:

σ = √Var(X)

Common Distributions #

Discrete Distributions #

Bernoulli Distribution:

  • Parameter: p (probability of success)
  • PMF: P(X = 1) = p, P(X = 0) = 1-p
  • Mean: p
  • Variance: p(1-p)

Binomial Distribution: Models the number of successes in n independent Bernoulli trials.

  • Parameters: n (number of trials), p (probability of success)
  • PMF: P(X = k) = C(n,k) × p^k × (1-p)^(n-k)
  • Mean: np
  • Variance: np(1-p)
  • MGF: M(t) = (pe^t + 1-p)^n

Geometric Distribution: Models the number of trials until the first success.

  • Parameter: p (probability of success)
  • PMF: P(X = k) = (1-p)^(k-1) × p, k = 1, 2, 3, …
  • Mean: 1/p
  • Variance: (1-p)/p²

Negative Binomial Distribution: Models the number of trials until the r-th success.

  • Parameters: r (number of successes), p (probability of success)
  • PMF: P(X = k) = C(k-1, r-1) × p^r × (1-p)^(k-r)
  • Mean: r/p
  • Variance: r(1-p)/p²

Poisson Distribution: Models the number of events in a fixed interval when events occur at a constant average rate.

  • Parameter: λ (average rate)
  • PMF: P(X = k) = (e^(-λ) × λ^k) / k!
  • Mean: λ
  • Variance: λ
  • MGF: M(t) = e^(λ(e^t - 1))

Continuous Distributions #

Uniform Distribution: All values in an interval are equally likely.

  • Parameters: a (minimum), b (maximum)
  • PDF: f(x) = 1/(b-a) for a ≤ x ≤ b, 0 elsewhere
  • CDF: F(x) = (x-a)/(b-a) for a ≤ x ≤ b
  • Mean: (a+b)/2
  • Variance: (b-a)²/12

Normal Distribution: The most important continuous distribution in statistics.

  • Parameters: μ (mean), σ² (variance)
  • PDF: f(x) = [1/(σ√(2π))] × exp[-(x-μ)²/(2σ²)]
  • Standard Normal: Z = (X-μ)/σ ~ N(0,1)
  • MGF: M(t) = exp(μt + σ²t²/2)

Exponential Distribution: Models time between events in a Poisson process.

  • Parameter: θ (mean) or λ = 1/θ (rate)
  • PDF: f(x) = (1/θ)e^(-x/θ) = λe^(-λx), x ≥ 0
  • CDF: F(x) = 1 - e^(-x/θ) = 1 - e^(-λx)
  • Mean: θ = 1/λ
  • Variance: θ² = 1/λ²
  • Memoryless Property: P(X > s+t | X > s) = P(X > t)

Gamma Distribution: Generalizes the exponential distribution.

  • Parameters: α (shape), θ (scale)
  • PDF: f(x) = [x^(α-1)e^(-x/θ)] / [θ^α Γ(α)], x ≥ 0
  • Mean: αθ
  • Variance: αθ²
  • Special case: When α = 1, it becomes exponential

Beta Distribution: Defined on [0,1], useful for modeling proportions.

  • Parameters: α, β (both > 0)
  • PDF: f(x) = [Γ(α+β)/(Γ(α)Γ(β))] × x^(α-1)(1-x)^(β-1), 0 ≤ x ≤ 1
  • Mean: α/(α+β)
  • Variance: αβ/[(α+β)²(α+β+1)]

Advanced Concepts #

Moment Generating Functions #

The MGF is a powerful tool for finding moments and identifying distributions.

Definition:

M_X(t) = E(e^(tX))

Properties:

  • If MGF exists, it uniquely determines the distribution
  • E(X^n) = M_X^(n)(0) (nth derivative evaluated at 0)
  • Mean: E(X) = M’_X(0)
  • Variance: Var(X) = M’’_X(0) - [M’_X(0)]²

MGF of Linear Transformations: If Y = aX + b, then M_Y(t) = e^(bt) × M_X(at)

Order Statistics #

When we have a random sample X₁, X₂, …, Xₙ, the order statistics are the values arranged in ascending order: X₍₁₎ ≤ X₍₂₎ ≤ … ≤ X₍ₙ₎

PDF of kth Order Statistic:

f_{X(k)}(x) = [n!/(k-1)!(n-k)!] × [F(x)]^(k-1) × [1-F(x)]^(n-k) × f(x)

Special Cases:

  • Minimum: f_{X(1)}(x) = n[1-F(x)]^(n-1)f(x)
  • Maximum: f_{X(n)}(x) = n[F(x)]^(n-1)f(x)

Joint Distributions #

For bivariate random variables (X,Y):

Joint PDF/PMF Properties:

  • f(x,y) ≥ 0
  • ∫∫ f(x,y)dxdy = 1 (or Σ Σ for discrete)

Marginal Distributions:

  • f_X(x) = ∫ f(x,y)dy
  • f_Y(y) = ∫ f(x,y)dx

Independence: X and Y are independent if f(x,y) = f_X(x) × f_Y(y)

Covariance:

Cov(X,Y) = E(XY) - E(X)E(Y)

Correlation:

ρ = Cov(X,Y)/[σ_X × σ_Y]

Transformation Techniques #

Method of Transformations: For Y = g(X) where g is monotonic:

f_Y(y) = f_X(g^(-1)(y)) × |dx/dy|

Jacobian Method: For transformations of multiple variables, use the Jacobian determinant.

MGF Method: If you can find the MGF of a transformation, you can identify the distribution.

Study Tips #

Time Management Strategies:

  • Allocate approximately 6 minutes per question
  • Practice identifying question types quickly
  • Learn when to skip difficult problems and return later
  • Use the elimination method for multiple-choice questions

Conceptual Understanding:

  • Master the relationships between different distributions
  • Understand when to apply each distribution type
  • Practice recognizing real-world scenarios that fit specific probability models
  • Focus on understanding derivations rather than pure memorization

Calculation Skills:

  • Become proficient with your BA-II Plus calculator
  • Practice common calculations like normal probabilities, combinations, and factorials
  • Learn shortcut methods for standard calculations
  • Master the use of probability tables and inverse functions

Practice Strategy:

  • Complete multiple full-length practice exams under timed conditions
  • Focus on areas where you consistently make mistakes
  • Practice problems from past SOA exams and study materials
  • Join study groups or online forums for additional practice problems

Formula Management:

  • Create a personal formula sheet for quick reference
  • Practice deriving key formulas from first principles
  • Understand the conditions under which each formula applies
  • Focus on formulas that appear frequently in practice problems

Exam Day Preparation #

Calculator Preparation:

  • Clear all memory and programs from your calculator
  • Practice using all necessary functions beforehand
  • Bring backup batteries (though CBT centers usually provide power)
  • Familiarize yourself with normal distribution functions and statistical calculations

Required Documentation:

  • Valid government-issued photo ID
  • Admission ticket (if applicable for your testing center)
  • Arrive at least 30 minutes before your scheduled exam time
  • Dress comfortably and appropriately for the testing environment

Mental Preparation:

  • Get adequate sleep the night before the exam
  • Eat a nutritious breakfast to maintain energy levels
  • Review key formulas and concepts briefly before the exam
  • Stay calm and confident in your preparation

During the Exam:

  • Read each question carefully and identify what is being asked
  • Check your work when time permits
  • Use the elimination method for difficult multiple-choice questions
  • Don’t spend too much time on any single question
  • Mark questions you’re unsure about and return to them if time allows

Final Words #

Success on SOA Exam P requires a combination of thorough conceptual understanding, consistent practice, and effective exam-taking strategies. This cheat sheet provides a comprehensive overview of the essential concepts, but remember that deep understanding comes through working numerous practice problems and applying these concepts in various scenarios.

The key to mastering probability theory is recognizing patterns in problem types and understanding when to apply specific distributions and techniques. Regular practice with timed exams will help you develop the speed and accuracy needed for success.

Remember that Exam P is not just about passing a test—it’s about building a foundation in probability theory that will serve you throughout your actuarial career. The concepts you learn here will be applied in more advanced exams and in your professional work as an actuary.

Stay focused, practice consistently, and approach the exam with confidence. Good luck with your preparation and exam success!