Exam P formula sheet — Probability
A free reference page for every formula on the SOA Exam P syllabus: probability rules, Bayes' theorem, all fourteen distributions with pdf, cdf, mean, variance and MGF, joint distributions, transformations and the central limit theorem.
29 pages, all free to read. Each one carries the formulas as they are actually written, a sketch of where they come from, a worked example whose answer is recomputed by the test suite, the memory hook that regenerates the formula when recall fails, and the specific traps that cost marks. 220 original practice questions drill this syllabus in the question bank.
“Exam P” is the descriptive name for the Society of Actuaries’ Probability exam and is used here nominatively. The official syllabus is at soa.org; this site is not affiliated with or endorsed by the SOA.
Probability foundations
Axioms and sets
Three axioms generate every counting-free probability result on the exam, and inclusion-exclusion is the one identity that appears in some form on almost every sitting.
Counting
Order matters or it does not, and repetition is allowed or it is not. Those two questions pick the formula every time.
Conditional probability
Conditioning shrinks the sample space to B and renormalises; independence is the special case where that shrinking changes nothing.
Bayes' theorem
Reverse a conditional probability: weight each cause by its prior and its likelihood, then normalise. The denominator is always the law of total probability.
Random variables
Random variables
The CDF is the one object every distribution has; densities, masses, survival functions and percentiles are all read off it.
Expectation and variance
E[X] is the centre, Var(X) the spread, and Var(X) = E[X²] − (E[X])² is the identity that turns most exam questions into two sums.
MGFs
M(t) = E[e^{tX}] encodes every moment in its derivatives at zero, and turns sums of independent variables into products.
Distributions
Discrete uniform
A finite set of equally likely integer outcomes, such as one fair die.
Binomial
The number of successes in a fixed number of independent trials that each succeed with the same probability.
Geometric
The number of failures before the first success in repeated independent trials.
Negative binomial
The number of failures before the rth success in repeated independent trials.
Hypergeometric
The number of successes in a sample drawn WITHOUT replacement from a finite population.
Poisson
The number of events in a fixed interval when events occur independently at a constant average rate.
Continuous uniform
A value equally likely to fall anywhere in an interval.
Exponential
The waiting time until the next event in a Poisson process; a memoryless lifetime.
Gamma
The waiting time until the αth event in a Poisson process; a flexible positive, right-skewed severity model.
Normal
A symmetric bell-shaped variable; the limiting distribution of standardised sums by the Central Limit Theorem.
Lognormal
A positive, right-skewed quantity whose logarithm is normal — the default model for claim severity and asset prices.
Beta
A proportion or probability constrained to [0, 1], with a shape set by two counts.
Chi-square
A sum of k squared independent standard normals; the sampling distribution of a variance.
Pareto
A heavy-tailed positive loss where large claims stay meaningfully likely.
Multivariate probability
Joint distributions
Integrate out what you do not care about to get a marginal; divide by a marginal to get a conditional. Drawing the region is most of the work.
Covariance
Covariance measures joint movement, correlation rescales it to [−1, 1], and the variance of a sum is where both actually get used.
Conditional expectation
Condition on what you do not know, then average it out. The double expectation and the conditional variance formula handle every compound model on the syllabus.
Applications
Transformations
Push a random variable through a function: use the CDF method when you can, the Jacobian when you must, and treat maxima and minima as CDF products.
Deductibles and limits
Insurance modifies a loss before it is paid. Each modification is an integral of the survival function over the relevant range.
Central limit theorem
Sums of many independent variables are approximately normal; standardise, look up Φ, and apply a continuity correction if the summands are integers.
Repeated trials
When each independent trial falls into one of several categories, the multinomial coefficient counts the arrangements and the probabilities multiply.
Mixtures
A population made of sub-populations is a mixture: probabilities average with the mixing weights, but variances do not.