Distributions - Exam P formula sheet

Every Exam P formula in distributions on one printable sheet, condensed from the 14 reference pages that teach them.

14 topics, 79 formula lines. Every line is condensed from the Exam P formula pages, which stay free to read in full. Print this page to get the sheet on paper; the site chrome drops away.

What is on this sheet

Distributions

Discrete uniform

Support: $x \in {a, a + 1, \dots, b}$
p(x): $p (x) = \frac{1}{b - a + 1}$
F(x): $F (x) = \frac{⌊ x ⌋ - a + 1}{b - a + 1}, a \leq x \leq b$
E[X]: $E [X] = \frac{a + b}{2}$
Var(X): $Var (X) = \frac{( b - a + 1 ) ^{2} - 1}{12}$
M(t): $M (t) = \frac{e ^{a t} ( e ^{n t} - 1 )}{n ( e ^{t} - 1 )}, n = b - a + 1$

Binomial

Support: $x \in {0, 1, \dots, n}$
p(x): $p (x) = (x n) p^{x} (1 - p)^{n - x}$
F(x): $F (x) = k = 0 \sum ⌊ x ⌋ (k n) p^{k} (1 - p)^{n - k}$ No shorter closed form. For large n the normal approximation with a continuity correction is the intended exam route.
E[X]: $E [X] = n p$
Var(X): $Var (X) = n p (1 - p)$
M(t): $M (t) = (1 - p + p e^{t})^{n}$

Geometric

Support: $x \in {0, 1, 2, \dots}$
p(x): $p (x) = (1 - p)^{x} p$
F(x): $F (x) = 1 - (1 - p)^{⌊ x ⌋ + 1}$
E[X]: $E [X] = \frac{1 - p}{p}$
Var(X): $Var (X) = \frac{1 - p}{p ^{2}}$
M(t): $M (t) = \frac{p}{1 - ( 1 - p ) e ^{t}}, t < - ln (1 - p)$

Negative binomial

Support: $x \in {0, 1, 2, \dots}$
p(x): $p (x) = (x x + r - 1) p^{r} (1 - p)^{x}$
F(x): $F (x) = I_{p} (r, ⌊ x ⌋ + 1)$ I is the regularized incomplete beta function; no elementary closed form exists.
E[X]: $E [X] = \frac{r ( 1 - p )}{p}$
Var(X): $Var (X) = \frac{r ( 1 - p )}{p ^{2}}$
M(t): $M (t) = (\frac{p}{1 - ( 1 - p ) e ^{t}})^{r}, t < - ln (1 - p)$

Hypergeometric

Support: $max (0, n - (N - K)) \leq x \leq min (n, K)$
p(x): $p (x) = \frac{( x K ) ( n - x N - K )}{( n N )}$
E[X]: $E [X] = n \frac{K}{N}$
Var(X): $Var (X) = n \frac{K}{N} (1 - \frac{K}{N}) \frac{N - n}{N - 1}$

Poisson

Support: $x \in {0, 1, 2, \dots}$
p(x): $p (x) = \frac{e ^{- λ} λ ^{x}}{x !}$
F(x): $F (x) = e^{- λ} k = 0 \sum ⌊ x ⌋ \frac{λ ^{k}}{k !}$
E[X]: $E [X] = λ$
Var(X): $Var (X) = λ$
M(t): $M (t) = e^{λ (e^{t} - 1)}$

Continuous uniform

Support: $a \leq x \leq b$
f(x): $f (x) = \frac{1}{b - a}$
F(x): $F (x) = \frac{x - a}{b - a}, a \leq x \leq b$
E[X]: $E [X] = \frac{a + b}{2}$
Var(X): $Var (X) = \frac{( b - a ) ^{2}}{12}$
M(t): $M (t) = \frac{e ^{t b} - e ^{t a}}{t ( b - a )}, t \neq = 0$

Exponential

Support: $x \geq 0$
f(x): $f (x) = λ e^{- λ x}$
F(x): $F (x) = 1 - e^{- λ x}$ The survival function S(x) = e^(−λx) is the form worth memorising - most questions ask for a tail.
E[X]: $E [X] = \frac{1}{λ}$
Var(X): $Var (X) = \frac{1}{λ ^{2}}$
M(t): $M (t) = \frac{λ}{λ - t}, t < λ$

Gamma

Support: $x > 0$
f(x): $f (x) = \frac{λ ^{α} x ^{α - 1} e ^{- λ x}}{Γ ( α )}$
F(x): $F (x) = \frac{γ ( α , λ x )}{Γ ( α )}$ Elementary only when α is a positive integer, where it collapses to a finite Poisson sum: F(x) = 1 − Σ e^(−λx)(λx)^k/k! for k = 0…α−1.
E[X]: $E [X] = \frac{α}{λ}$
Var(X): $Var (X) = \frac{α}{λ ^{2}}$
M(t): $M (t) = (\frac{λ}{λ - t})^{α}, t < λ$

Normal

Support: $- \infty < x < \infty$
f(x): $f (x) = \frac{1}{σ 2 π} exp (- \frac{( x - μ ) ^{2}}{2 σ ^{2}})$
F(x): $F (x) = Φ (\frac{x - μ}{σ})$ No elementary antiderivative - Φ is read from the standard normal table.
E[X]: $E [X] = μ$
Var(X): $Var (X) = σ^{2}$
M(t): $M (t) = exp (μ t + \frac{σ ^{2} t ^{2}}{2})$

Lognormal

Support: $x > 0$
f(x): $f (x) = \frac{1}{x σ 2 π} exp (- \frac{( ln x - μ ) ^{2}}{2 σ ^{2}})$
F(x): $F (x) = Φ (\frac{ln x - μ}{σ})$
E[X]: $E [X] = e^{μ + σ^{2} /2}$
Var(X): $Var (X) = (e^{σ^{2}} - 1) e^{2 μ + σ^{2}}$

Beta

Support: $0 < x < 1$
f(x): $f (x) = \frac{Γ ( α + β )}{Γ ( α ) Γ ( β )} x^{α - 1} (1 - x)^{β - 1}$
F(x): $F (x) = I_{x} (α, β)$ The regularized incomplete beta function. Elementary only when α or β is a small positive integer.
E[X]: $E [X] = \frac{α}{α + β}$
Var(X): $Var (X) = \frac{α β}{( α + β ) ^{2} ( α + β + 1 )}$

Chi-square

Support: $x > 0$
f(x): $f (x) = \frac{x ^{k /2 - 1} e ^{- x /2}}{2 ^{k /2} Γ ( k /2 )}$
F(x): $F (x) = \frac{γ ( k /2 , x /2 )}{Γ ( k /2 )}$
E[X]: $E [X] = k$
Var(X): $Var (X) = 2 k$
M(t): $M (t) = (1 - 2 t)^{- k /2}, t < \frac{1}{2}$

Pareto

Support: $x > 0$
f(x): $f (x) = \frac{α θ ^{α}}{( x + θ ) ^{α + 1}}$
F(x): $F (x) = 1 - (\frac{θ}{x + θ})^{α}$
E[X]: $E [X] = \frac{θ}{α - 1}, α > 1$
Var(X): $Var (X) = \frac{α θ ^{2}}{( α - 1 ) ^{2} ( α - 2 )}, α > 2$

Distributions - Exam P formula sheet. Formulas condensed from the free reference pages on this site. Not affiliated with or endorsed by the Society of Actuaries.