FRM Part I · FRM Exam Part I

Fundamentals of Probability: formula sheet

Full chapter guide

Key formulas

PMF conditions: 0 ≤ p(x) ≤ 1 and Σ p(x) = 1; p(x) = P(X = x). Applies to discrete variables only.
PDF conditions: f(x) ≥ 0 and ∫ f(x) dx = 1 over all x; f(x) is a density, not a probability. It can exceed 1.
Probability from a PDF: P(a ≤ X ≤ b) = ∫ from a to b of f(x) dx; For continuous X, P(X = a) = 0, so including or excluding endpoints does not matter.
CDF definition: F(x) = P(X ≤ x); Discrete: sum of p over values ≤ x. Continuous: integral of f up to x.
Interval probability from CDF: P(a < X ≤ b) = F(b) − F(a); For discrete X, endpoint a is excluded and b is included, so check the inequality signs.
Upper tail: P(X > x) = 1 − F(x); Used for loss exceedance and tail questions.
PDF from CDF: f(x) = dF(x)/dx; For continuous variables where F is differentiable.
Quantile: x_α = F⁻¹(α), where F(x_α) = α; VaR at confidence c is the quantile of the loss distribution at α = c.
Complement rule: P(Aᶜ) = 1 − P(A); Use it when 'at least one' is easier as 1 minus 'none'.
Addition rule (general): P(A ∪ B) = P(A) + P(B) − P(A ∩ B); Works for any two events. Subtract the overlap so it is not counted twice.
Addition rule (mutually exclusive): P(A ∪ B) = P(A) + P(B); Valid only when P(A ∩ B) = 0.
Mutually exclusive events: P(A ∩ B) = 0; They cannot occur together.
Exhaustive events: P(A₁ ∪ A₂ ∪ … ∪ Aₙ) = 1; Together they cover the sample space.
Partition: P(A₁) + P(A₂) + … + P(Aₙ) = 1; Holds when the events are both mutually exclusive and exhaustive.
Independence: P(A ∩ B) = P(A) × P(B); A different idea from exclusivity.
Union of three events: P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A∩B) − P(A∩C) − P(B∩C) + P(A∩B∩C); Add singles, subtract pairs, add the triple.
Conditional probability: P(A | B) = P(A ∩ B) ÷ P(B); Valid only when P(B) > 0.
Multiplication rule: P(A ∩ B) = P(A | B) × P(B) = P(B | A) × P(A); Works for any events, dependent or not.
Independence: P(A ∩ B) = P(A) × P(B), equivalently P(A | B) = P(A); For events with non-zero probability, any one of these conditions implies the others.
Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); If mutually exclusive, the last term is 0.
Law of total probability: P(A) = Σ P(A | Bᵢ) × P(Bᵢ); The Bᵢ must be mutually exclusive and exhaustive.
Mutually exclusive events: P(A ∩ B) = 0; Two mutually exclusive events with positive probabilities are not independent.
Conditional probability: P(A | B) = P(A ∩ B) ÷ P(B); Valid when P(B) > 0. Rearranged: P(A ∩ B) = P(B | A) × P(A).
Bayes' theorem (two events): P(A | B) = P(B | A) × P(A) ÷ P(B); A is the hypothesis, B is the evidence. Needs P(B) > 0.
Law of total probability: P(B) = P(B | A) × P(A) + P(B | not A) × P(not A); Works because A and not A are mutually exclusive and exhaustive.
Bayes' theorem (several states): P(Aᵢ | B) = P(B | Aᵢ) × P(Aᵢ) ÷ Σⱼ [P(B | Aⱼ) × P(Aⱼ)]; The Aᵢ must be mutually exclusive and exhaustive. Posteriors sum to 1.
Odds form: Posterior odds = prior odds × likelihood ratio; Odds = p ÷ (1 − p). Likelihood ratio = P(B | A) ÷ P(B | not A). Convert odds back with p = odds ÷ (1 + odds).
Expected value (discrete): E(X) = Σ xᵢ · p(xᵢ); Probabilities must sum to 1. For a continuous variable, replace the sum with an integral of x·f(x).
Variance (definition): Var(X) = σ² = E[(X − μ)²] = Σ (xᵢ − μ)² · p(xᵢ); Use this when the mean is a clean number.
Variance (shortcut): Var(X) = E(X²) − [E(X)]²; E(X²) = Σ xᵢ² · p(xᵢ). Usually faster with a calculator. Result can never be negative.
Standard deviation: σ = √Var(X); Same units as X. Not the same as variance.
Linear transformation of the mean: E(a + bX) = a + b·E(X); Holds for any constants a and b.
Linear transformation of the variance: Var(a + bX) = b² · Var(X); The constant a drops out. Standard deviation of a + bX is |b| · σ.
Expectation of a sum: E(aX + bY) = a·E(X) + b·E(Y); Needs no independence assumption.
Expectation of a function: E[g(X)] = Σ g(xᵢ) · p(xᵢ); In general E[g(X)] ≠ g(E[X]), for example E(X²) ≠ [E(X)]² unless the variance is zero.
Covariance: Cov(X,Y) = E[(X − μX)(Y − μY)] = E[XY] − E[X]E[Y]; Cov(X,X) = Var(X).
Correlation: ρ = Cov(X,Y) ÷ (σX σY); Always between −1 and +1. Unitless.
Variance of a sum: Var(aX + bY) = a²σX² + b²σY² + 2ab·Cov(X,Y); Use −2ab·Cov for a difference X − Y with a = b = 1: Var(X − Y) = σX² + σY² − 2Cov.
Covariance scaling: Cov(aX + c, bY + d) = ab·Cov(X,Y); Constants added do not change covariance.
Raw moment: k-th raw moment = E[X^k]; Taken about zero.
Central moment: k-th central moment = E[(X − μ)^k]; Taken about the mean. Second central moment is variance.
Skewness: Skew = E[(X − μ)³] ÷ σ³; Zero for symmetric distributions.
Kurtosis: Kurt = E[(X − μ)⁴] ÷ σ⁴; Excess kurtosis = Kurt − 3; Normal has kurtosis 3.
Independence and covariance: Independent ⇒ Cov = 0, but Cov = 0 does not imply independent; Zero covariance only rules out linear dependence.

Quick revision

A probability is between 0 and 1, and all outcomes of a distribution sum (or integrate) to 1.
The CDF F(x) = P(X ≤ x) never decreases as x rises.
For any two events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
Mutually exclusive events cannot occur together, so P(A ∩ B) = 0.
Exhaustive events together cover the whole sample space.
Conditional probability: P(A | B) = P(A ∩ B) ÷ P(B), for P(B) > 0.
Events are independent if P(A ∩ B) = P(A) × P(B), which means P(A | B) = P(A) when P(B) > 0.
Bayes: P(A | B) = P(B | A) × P(A) ÷ P(B), with P(B) from the law of total probability.
E(X) = Σ x × P(x) for a discrete variable; Var(X) = E(X²) − [E(X)]².
Var(aX + b) = a² × Var(X); the constant b does not change variance.
Cov(X, Y) = E(XY) − E(X)E(Y); Corr = Cov ÷ (σX × σY), and it lies between −1 and +1.
Independent variables have zero covariance, but zero covariance does not by itself prove independence.

Common mistakes

Treating the PDF value f(x) as a probability. Fix: For continuous variables, probability is always an area. Only the PMF value at a point is a probability.
Saying P(X = x) is positive for a continuous variable. Fix: For a continuous variable, P(X = x) = 0. Only intervals have positive probability.
Treating mutually exclusive as independent Fix: Exclusive means P(A ∩ B) = 0. Independent means P(A ∩ B) = P(A) P(B). With non-zero probabilities, exclusive events are dependent.
Adding P(A) + P(B) for events that overlap Fix: Use the general addition rule and subtract P(A ∩ B) unless the events are exclusive.
Dividing by the wrong probability, giving P(B | A) instead of P(A | B). Fix: The condition is always the denominator. Write the event after 'given' first.
Treating mutually exclusive events as independent. Fix: Mutually exclusive means P(A ∩ B) = 0. Independent means the joint equals the product. With positive probabilities they cannot both hold.
Treating P(B | A) as if it were P(A | B) Fix: Write each probability with its condition explicitly. Ask which event is given and which is being asked about.
Ignoring the prior (base rate) Fix: Always start by writing P(A). A low prior pulls the posterior down, even with a strong likelihood.
Forgetting to square b when finding Var(a + bX). Fix: Variance is in squared units, so the multiplier is b². Standard deviation uses |b|.
Adding the constant a to the variance. Fix: A constant shift does not change spread. Var(a + bX) has no a term.

Exam tips

First identify discrete or continuous. Many wrong options come from using the wrong function.
For discrete CDF questions, read the inequality signs carefully. This is where marks are lost.
For simple PDFs, use geometry for area instead of integration. It is faster and less error-prone.
Link CDF questions to VaR: a 95% VaR is the loss level x where F(x) = 0.95.
If a PMF or PDF has an unknown constant, solve it first using total probability = 1.
Read the wording: 'or' means union, 'and' means intersection, 'not' means complement.
If the question says exclusive, drop the overlap term. If it does not say so, do not assume it.
Use the complement for any 'at least one' question.