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Free SOA Exam P Formula Sheet PDF, No Signup (2026)

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All Exam P Formulas

Probability Fundamentals 5 items

Set complement

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

Addition rule (two events)

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Addition rule (three events)

P(A \cup B \cup C) = P(A)+P(B)+P(C) - P(A\cap B) - P(A\cap C) - P(B\cap C) + P(A\cap B\cap C)

Permutations (ordered)

_nP_r = \dfrac{n!}{(n-r)!}

n

objects taken

r

at a time, order matters

Combinations (unordered)

\binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}

n

objects taken

r

at a time, order does not matter

Conditional Probability & Bayes 5 items

Conditional probability

P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}, \quad P(B)>0

Multiplication rule

P(A \cap B) = P(A \mid B)\,P(B) = P(B \mid A)\,P(A)

Law of total probability

P(A) = \sum_{i} P(A \mid B_i)\,P(B_i)

where

\{B_i\}

is a partition of the sample space

Bayes' theorem

P(B_i \mid A) = \dfrac{P(A \mid B_i)\,P(B_i)}{\sum_j P(A \mid B_j)\,P(B_j)}

Independence of two events

A

and

B

are independent iff

P(A \cap B) = P(A)\,P(B)

(equivalently:

P(A\mid B)=P(A)

)

Discrete Distributions 6 items

Bernoulli distribution — PMF, mean, variance

P(X=1)=p,\; P(X=0)=1-p

E[X]=p,\quad \text{Var}(X)=p(1-p)

Binomial distribution — PMF, mean, variance

P(X=k)=(nk)pk(1−p)n−kP(X=k)=\binom{n}{k}p^k(1-p)^{n-k}P(X=k)=(kn​)pk(1−p)n−k
E[X]=np,Var(X)=np(1−p)E[X]=np,\quad \text{Var}(X)=np(1-p)E[X]=np,Var(X)=np(1−p)
k=0,1,…,nk=0,1,\ldots,nk=0,1,…,n

Poisson distribution — PMF, mean, variance

P(X=k)=e−λλkk!P(X=k)=\dfrac{e^{-\lambda}\lambda^k}{k!}P(X=k)=k!e−λλk​
E[X]=λ,Var(X)=λE[X]=\lambda,\quad \text{Var}(X)=\lambdaE[X]=λ,Var(X)=λ
k=0,1,2,…k=0,1,2,\ldotsk=0,1,2,…

Geometric distribution — PMF, mean, variance

P(X=k)=(1−p)k−1pP(X=k)=(1-p)^{k-1}pP(X=k)=(1−p)k−1p (number of trials to first success)
E[X]=1p,Var(X)=1−pp2E[X]=\dfrac{1}{p},\quad \text{Var}(X)=\dfrac{1-p}{p^2}E[X]=p1​,Var(X)=p21−p​
k=1,2,…k=1,2,\ldotsk=1,2,…

Negative Binomial distribution — PMF, mean, variance

P(X=k)=\binom{k-1}{r-1}p^r(1-p)^{k-r}

(trials for

r

th success)

E[X]=\dfrac{r}{p},\quad \text{Var}(X)=\dfrac{r(1-p)}{p^2}

Hypergeometric distribution — PMF, mean, variance

P(X=k)=\dfrac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}

E[X]=\dfrac{nK}{N},\quad \text{Var}(X)=\dfrac{nK(N-K)(N-n)}{N^2(N-1)}

Continuous Distributions 8 items

Uniform distribution — PDF, mean, variance

f(x)=\dfrac{1}{b-a},\quad a<x<b

E[X]=\dfrac{a+b}{2},\quad \text{Var}(X)=\dfrac{(b-a)^2}{12}

Exponential distribution — PDF, CDF, mean, variance

f(x)=\lambda e^{-\lambda x},\quad F(x)=1-e^{-\lambda x},\quad x>0

E[X]=\dfrac{1}{\lambda},\quad \text{Var}(X)=\dfrac{1}{\lambda^2}

Memoryless property (Exponential / Geometric)

P(X > s+t \mid X > s) = P(X > t)

Only the Exponential (continuous) and Geometric (discrete) satisfy this.

Normal distribution — PDF

f(x)=\dfrac{1}{\sigma\sqrt{2\pi}}\exp\!\left(-\dfrac{(x-\mu)^2}{2\sigma^2}\right)

E[X]=\mu,\quad \text{Var}(X)=\sigma^2

Gamma distribution — PDF, mean, variance

f(x)=xα−1e−x/θΓ(α) θα,x>0f(x)=\dfrac{x^{\alpha-1}e^{-x/\theta}}{\Gamma(\alpha)\,\theta^\alpha},\quad x>0f(x)=Γ(α)θαxα−1e−x/θ​,x>0
E[X]=αθ,Var(X)=αθ2E[X]=\alpha\theta,\quad \text{Var}(X)=\alpha\theta^2E[X]=αθ,Var(X)=αθ2
α\alphaα=shape, θ\thetaθ=scale

Weibull distribution — PDF, mean

f(x)=\dfrac{\tau}{\theta}\left(\dfrac{x}{\theta}\right)^{\tau-1}e^{-(x/\theta)^\tau},\quad x>0

E[X]=\theta\,\Gamma\!\left(1+\tfrac{1}{\tau}\right)

Lognormal distribution — PDF, mean, variance

f(x)=\dfrac{1}{x\sigma\sqrt{2\pi}}\exp\!\left(-\dfrac{(\ln x-\mu)^2}{2\sigma^2}\right),\quad x>0

E[X]=e^{\mu+\sigma^2/2},\quad \text{Var}(X)=e^{2\mu+\sigma^2}(e^{\sigma^2}-1)

Beta distribution — PDF, mean

f(x)=\dfrac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)},\quad 0<x<1

E[X]=\dfrac{\alpha}{\alpha+\beta},\quad \text{Var}(X)=\dfrac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}

Joint & Marginal Distributions 3 items

Joint PDF — marginal densities

f_X(x)=\int_{-\infty}^{\infty}f_{X,Y}(x,y)\,dy

f_Y(y)=\int_{-\infty}^{\infty}f_{X,Y}(x,y)\,dx

Conditional PDF

f_{Y\mid X}(y\mid x)=\dfrac{f_{X,Y}(x,y)}{f_X(x)}

Independence of continuous RVs

X,Y

independent iff

f_{X,Y}(x,y)=f_X(x)\,f_Y(y)

for all

x,y

Expectation & Variance 7 items

MGF definition and moment extraction

M_X(t)=E[e^{tX}]

E[X^n]=M_X^{(n)}(0)

(

n

th derivative at

t=0

)

Variance shortcut

\text{Var}(X)=E[X^2]-(E[X])^2

Covariance

\text{Cov}(X,Y)=E[XY]-E[X]\,E[Y]

X,Y

independent:

\text{Cov}(X,Y)=0

Correlation coefficient

\rho_{XY}=\dfrac{\text{Cov}(X,Y)}{\sigma_X\,\sigma_Y},\quad -1\le\rho\le1

Variance of a linear combination

\text{Var}(aX+bY)=a^2\text{Var}(X)+b^2\text{Var}(Y)+2ab\,\text{Cov}(X,Y)

Law of total expectation

E[X]=E[E[X\mid Y]]

Law of total variance

\text{Var}(X)=E[\text{Var}(X\mid Y)]+\text{Var}(E[X\mid Y])

Insurance Applications 5 items

Ordinary deductible — payment per loss

YL=max⁡(X−d, 0)Y^L = \max(X-d,\,0)YL=max(X−d,0)
E[YL]=E[X]−E[X∧d]E[Y^L]=E[X]-E[X\wedge d]E[YL]=E[X]−E[X∧d]
where ddd = deductible, XXX = ground-up loss

Limited expected value (LEV)

E[X\wedge u]=\int_0^u S(x)\,dx = \int_0^u [1-F(x)]\,dx

for

X\ge 0

Payment per payment (excess loss)

e(d)=E[X-d\mid X>d]=\dfrac{E[X]-E[X\wedge d]}{1-F(d)}

d

= deductible

Policy limit — payment per loss

YL=min⁡(X, u)=X∧uY^L=\min(X,\,u)=X\wedge uYL=min(X,u)=X∧u
E[YL]=E[X∧u]E[Y^L]=E[X\wedge u]E[YL]=E[X∧u]
uuu = policy limit

Stop-loss (aggregate) premium

E[(S-d)_+]=E[S]-E[S\wedge d]

S

= aggregate loss,

d

= retention (aggregate deductible)

Topic 0 28 items

De Morgan's Laws

(A \cup B)^c = A^c \cap B^c

and

(A \cap B)^c = A^c \cup B^c

; A, B = events,

^c

= complement,

\cup

= union,

\cap

= intersection

Stars and bars distribution count

\binom{n+k-1}{k-1}

, n = identical objects distributed, k = distinct bins (non-negative integer solutions to

x_1+\cdots+x_k=n

); use

\binom{n-1}{k-1}

if each bin needs at least one

Addition rule for mutually exclusive events

P(A \cup B) = P(A) + P(B)

when

P(A \cap B) = 0

; A, B = mutually exclusive (disjoint) events that cannot both occur

Feasibility bounds for an intersection probability

\max(0, P(A)+P(B)-1) \leq P(A \cap B) \leq \min(P(A), P(B))

, lower bound = Bonferroni, upper bound from subset containment

Chain rule for the probability of an intersection of events

P(A_1 \cap \cdots \cap A_n) = P(A_1)P(A_2 \mid A_1)P(A_3 \mid A_1 \cap A_2)\cdots P(A_n \mid A_1 \cap \cdots \cap A_{n-1})

, each factor conditions on all prior events

Odds form of Bayes' theorem

\frac{P(B \mid A)}{P(B^c \mid A)} = \frac{P(A \mid B)}{P(A \mid B^c)} \cdot \frac{P(B)}{P(B^c)}

, posterior odds = likelihood ratio × prior odds; A = evidence, B = hypothesis, B^c = complement

Distinguishable permutations of a multiset

N = \dfrac{n!}{n_1!\,n_2!\cdots n_k!}

, n = total items,

n_i

= count of identical items of type i, k = number of distinct types

Survival function in terms of the CDF

S_X(x) = P(X > x) = 1 - F_X(x)

S_X

= survival function,

F_X(x)=P(X\le x)

= CDF, x = threshold value

Conditional CDF of a left-truncated distribution

F_{X|X>a}(x) = \frac{F_X(x) - F_X(a)}{1 - F_X(a)}, \ x > a

, F = CDF, a = truncation point, x = value

Law of the Unconscious Statistician (LOTUS)

E[g(X)] = \int_{-\infty}^{\infty} g(x) f_X(x)\, dx

, g = function of X,

f_X

= pdf of X (use sum with

p_X

(x) if discrete)

Chebyshev's inequality

P(|X - \mu| \geq k\sigma) \leq \frac{1}{k^2}

, μ = mean, σ = standard deviation, k = number of SDs; equivalently at least

1 - 1/k^2

lies within k SDs

Expected payment per loss with deductible and benefit limit

E[Y^L] = E[X \wedge (d+u)] - E[X \wedge d]

, d = deductible, u = benefit limit, X = ground-up loss,

X \wedge a

= loss capped at a

Second moment of the excess loss for an exponential

E[((X-d)_+)^2] = S(d)\cdot 2\theta^2 = e^{-d/\theta}\cdot 2\theta^2

, d = deductible, θ = exponential mean, S(d) = survival function at d

Pareto distribution pdf, mean, and variance

f(x) = \frac{\alpha\theta^\alpha}{(x+\theta)^{\alpha+1}}

E[X]=\frac{\theta}{\alpha-1}

(

\alpha>1

\text{Var}(X)=\frac{\alpha\theta^2}{(\alpha-1)^2(\alpha-2)}

(

\alpha>2

); α = shape, θ = scale, x > 0

Finite-population correction factor for hypergeometric variance

\text{Var}(X) = n\cdot\frac{K}{N}\cdot\frac{N-K}{N}\cdot\frac{N-n}{N-1}

, n = draws, N = population, K = successes in population, X = successes drawn

Sum of independent Poisson random variables

\text{Poisson}(\lambda_1) + \text{Poisson}(\lambda_2) = \text{Poisson}(\lambda_1 + \lambda_2)

, λ₁, λ₂ = rates of independent Poisson counts; combined count is Poisson with summed rate

Discrete uniform distribution pmf mean and variance

P(X=x)=\tfrac{1}{k},\ E[X]=\tfrac{a+b}{2},\ \text{Var}(X)=\tfrac{k^2-1}{12}

; a, b = min, max integers; k = b-a+1 = count of values; x in {a,...,b}

Gamma-Poisson tail connection for integer shape

P(X \le t) = P(\text{Poisson}(t/\theta) \ge \alpha)

, equivalently

P(X > t) = P(\text{Poisson}(t/\theta) \le \alpha - 1)

; X~Gamma(α,θ), α = integer shape, θ = scale, t = time

Beta distribution variance

\text{Var}(X) = \frac{ab}{(a+b)^2(a+b+1)}

, a, b = shape parameters of Beta on (0,1)

Rectangular probability from a joint CDF

P(a < X \leq b,\ c < Y \leq d) = F(b,d) - F(a,d) - F(b,c) + F(a,c)

, F = joint CDF; a,b = X limits; c,d = Y limits

Conditional PMF of a discrete random variable

p_{Y|X}(y|x) = \dfrac{p_{X,Y}(x,y)}{p_X(x)}

p_{X,Y}

= joint PMF,

p_X

= marginal PMF of X, requires

p_X(x) > 0

Compound mean (Wald's identity for the mean)

E[S] = E[N] \cdot E[X]

, S = aggregate loss, N = random claim count, X = i.i.d. severity independent of N

Variance of a compound random sum

\text{Var}(S) = E[N]\,\text{Var}(X) + \text{Var}(N)\,(E[X])^2

, S = sum of N losses, N = random claim count, X = individual loss severity

Covariance via conditional expectation

\text{Cov}(X,Y) = E[X \cdot E[Y|X]] - E[X] \cdot E[Y]

, E[Y|X] = conditional mean of Y given X, E[X], E[Y] = marginal means

PDF of the k-th order statistic

f_{X_{(k)}}(x) = \frac{n!}{(k-1)!(n-k)!}[F(x)]^{k-1}[1-F(x)]^{n-k}f(x)

, n = sample size, k = rank, F = cdf, f = pdf

Linear combination of independent normal random variables

aX + bY + c \sim N(a\mu_X + b\mu_Y + c,\; a^2\sigma_X^2 + b^2\sigma_Y^2)

, a, b, c = constants, μ = mean, σ² = variance; X, Y independent normals

Mean of a linear combination of random variables

E\left[\sum_{i=1}^n a_i X_i + b\right] = \sum_{i=1}^n a_i E[X_i] + b

a_i

= constant coefficients,

X_i

= random variables, b = additive constant (no independence needed)

Central Limit Theorem normal approximation for a sum

S_n = \sum_{i=1}^n X_i \approx N(n\mu, n\sigma^2)

, standardize

z = \frac{s - n\mu}{\sigma\sqrt{n}}

;

\mu

= mean,

\sigma^2

= variance, n = number of iid terms

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