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Distributions - Exam P formula sheet Every Exam P formula in distributions on one printable sheet, condensed from the 14 reference pages that teach them.
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Distributions Support x ∈ { a , a + 1 , … , b } x \in \{a,\, a+1,\, \dots,\, b\} x ∈ { a , a + 1 , … , b } p(x) p ( x ) = 1 b − a + 1 p(x) = \dfrac{1}{b-a+1} p ( x ) = b − a + 1 1 F(x) F ( x ) = ⌊ x ⌋ − a + 1 b − a + 1 , a ≤ x ≤ b F(x) = \dfrac{\lfloor x \rfloor - a + 1}{b-a+1}, \quad a \le x \le b F ( x ) = b − a + 1 ⌊ x ⌋ − a + 1 , a ≤ x ≤ b E[X] E [ X ] = a + b 2 E[X] = \dfrac{a+b}{2} E [ X ] = 2 a + b Var(X) Var ( X ) = ( b − a + 1 ) 2 − 1 12 \operatorname{Var}(X) = \dfrac{(b-a+1)^2 - 1}{12} Var ( X ) = 12 ( b − a + 1 ) 2 − 1 M(t) M ( t ) = e a t ( e n t − 1 ) n ( e t − 1 ) , n = b − a + 1 M(t) = \dfrac{e^{at}\left(e^{n t} - 1\right)}{n\left(e^{t} - 1\right)}, \quad n = b-a+1 M ( t ) = n ( e t − 1 ) e a t ( e n t − 1 ) , n = b − a + 1 Support x ∈ { 0 , 1 , … , n } x \in \{0, 1, \dots, n\} x ∈ { 0 , 1 , … , n } p(x) p ( x ) = ( n x ) p x ( 1 − p ) n − x p(x) = \binom{n}{x} p^{x} (1-p)^{n-x} p ( x ) = ( x n ) p x ( 1 − p ) n − x F(x) F ( x ) = ∑ k = 0 ⌊ x ⌋ ( n k ) p k ( 1 − p ) n − k F(x) = \displaystyle\sum_{k=0}^{\lfloor x \rfloor} \binom{n}{k} p^{k}(1-p)^{n-k} F ( x ) = k = 0 ∑ ⌊ 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 E[X] = np E [ X ] = n p Var(X) Var ( X ) = n p ( 1 − p ) \operatorname{Var}(X) = np(1-p) Var ( X ) = n p ( 1 − p ) M(t) M ( t ) = ( 1 − p + p e t ) n M(t) = \left(1 - p + p e^{t}\right)^{n} M ( t ) = ( 1 − p + p e t ) n Support x ∈ { 0 , 1 , 2 , … } x \in \{0, 1, 2, \dots\} x ∈ { 0 , 1 , 2 , … } p(x) p ( x ) = ( 1 − p ) x p p(x) = (1-p)^{x} p p ( x ) = ( 1 − p ) x p F(x) F ( x ) = 1 − ( 1 − p ) ⌊ x ⌋ + 1 F(x) = 1 - (1-p)^{\lfloor x \rfloor + 1} F ( x ) = 1 − ( 1 − p ) ⌊ x ⌋ + 1 E[X] E [ X ] = 1 − p p E[X] = \dfrac{1-p}{p} E [ X ] = p 1 − p Var(X) Var ( X ) = 1 − p p 2 \operatorname{Var}(X) = \dfrac{1-p}{p^{2}} Var ( X ) = p 2 1 − p M(t) M ( t ) = p 1 − ( 1 − p ) e t , t < − ln ( 1 − p ) M(t) = \dfrac{p}{1 - (1-p)e^{t}}, \quad t < -\ln(1-p) M ( t ) = 1 − ( 1 − p ) e t p , t < − ln ( 1 − p ) Support x ∈ { 0 , 1 , 2 , … } x \in \{0, 1, 2, \dots\} x ∈ { 0 , 1 , 2 , … } p(x) p ( x ) = ( x + r − 1 x ) p r ( 1 − p ) x p(x) = \binom{x + r - 1}{x} p^{r} (1-p)^{x} p ( x ) = ( x x + r − 1 ) p r ( 1 − p ) x F(x) F ( x ) = I p ( r , ⌊ x ⌋ + 1 ) F(x) = I_{p}\!\left(r,\, \lfloor x \rfloor + 1\right) F ( x ) = I p ( r , ⌊ x ⌋ + 1 ) I is the regularized incomplete beta function; no elementary closed form exists. E[X] E [ X ] = r ( 1 − p ) p E[X] = \dfrac{r(1-p)}{p} E [ X ] = p r ( 1 − p ) Var(X) Var ( X ) = r ( 1 − p ) p 2 \operatorname{Var}(X) = \dfrac{r(1-p)}{p^{2}} Var ( X ) = p 2 r ( 1 − p ) M(t) M ( t ) = ( p 1 − ( 1 − p ) e t ) r , t < − ln ( 1 − p ) M(t) = \left(\dfrac{p}{1 - (1-p)e^{t}}\right)^{r}, \quad t < -\ln(1-p) M ( t ) = ( 1 − ( 1 − p ) e t p ) r , t < − ln ( 1 − p ) Support max ( 0 , n − ( N − K ) ) ≤ x ≤ min ( n , K ) \max(0,\, n-(N-K)) \le x \le \min(n,\, K) max ( 0 , n − ( N − K )) ≤ x ≤ min ( n , K ) p(x) p ( x ) = ( K x ) ( N − K n − x ) ( N n ) p(x) = \dfrac{\dbinom{K}{x}\dbinom{N-K}{n-x}}{\dbinom{N}{n}} p ( x ) = ( n N ) ( x K ) ( n − x N − K ) E[X] E [ X ] = n K N E[X] = n\dfrac{K}{N} E [ X ] = n N K Var(X) Var ( X ) = n K N ( 1 − K N ) N − n N − 1 \operatorname{Var}(X) = n\dfrac{K}{N}\left(1-\dfrac{K}{N}\right)\dfrac{N-n}{N-1} Var ( X ) = n N K ( 1 − N K ) N − 1 N − n Support x ∈ { 0 , 1 , 2 , … } x \in \{0, 1, 2, \dots\} x ∈ { 0 , 1 , 2 , … } p(x) p ( x ) = e − λ λ x x ! p(x) = \dfrac{e^{-\lambda}\lambda^{x}}{x!} p ( x ) = x ! e − λ λ x F(x) F ( x ) = e − λ ∑ k = 0 ⌊ x ⌋ λ k k ! F(x) = e^{-\lambda}\displaystyle\sum_{k=0}^{\lfloor x \rfloor} \dfrac{\lambda^{k}}{k!} F ( x ) = e − λ k = 0 ∑ ⌊ x ⌋ k ! λ k E[X] E [ X ] = λ E[X] = \lambda E [ X ] = λ Var(X) Var ( X ) = λ \operatorname{Var}(X) = \lambda Var ( X ) = λ M(t) M ( t ) = e λ ( e t − 1 ) M(t) = e^{\lambda\left(e^{t}-1\right)} M ( t ) = e λ ( e t − 1 ) Support a ≤ x ≤ b a \le x \le b a ≤ x ≤ b f(x) f ( x ) = 1 b − a f(x) = \dfrac{1}{b-a} f ( x ) = b − a 1 F(x) F ( x ) = x − a b − a , a ≤ x ≤ b F(x) = \dfrac{x-a}{b-a}, \quad a \le x \le b F ( x ) = b − a x − a , a ≤ x ≤ b E[X] E [ X ] = a + b 2 E[X] = \dfrac{a+b}{2} E [ X ] = 2 a + b Var(X) Var ( X ) = ( b − a ) 2 12 \operatorname{Var}(X) = \dfrac{(b-a)^{2}}{12} Var ( X ) = 12 ( b − a ) 2 M(t) M ( t ) = e t b − e t a t ( b − a ) , t ≠ 0 M(t) = \dfrac{e^{tb} - e^{ta}}{t(b-a)}, \quad t \ne 0 M ( t ) = t ( b − a ) e t b − e t a , t = 0 f(x) f ( x ) = λ e − λ x f(x) = \lambda e^{-\lambda x} f ( x ) = λ e − λ x F(x) F ( x ) = 1 − e − λ x F(x) = 1 - e^{-\lambda 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 ] = 1 λ E[X] = \dfrac{1}{\lambda} E [ X ] = λ 1 Var(X) Var ( X ) = 1 λ 2 \operatorname{Var}(X) = \dfrac{1}{\lambda^{2}} Var ( X ) = λ 2 1 M(t) M ( t ) = λ λ − t , t < λ M(t) = \dfrac{\lambda}{\lambda - t}, \quad t < \lambda M ( t ) = λ − t λ , t < λ f(x) f ( x ) = λ α x α − 1 e − λ x Γ ( α ) f(x) = \dfrac{\lambda^{\alpha} x^{\alpha - 1} e^{-\lambda x}}{\Gamma(\alpha)} f ( x ) = Γ ( α ) λ α x α − 1 e − λ x F(x) F ( x ) = γ ( α , λ x ) Γ ( α ) F(x) = \dfrac{\gamma(\alpha,\, \lambda x)}{\Gamma(\alpha)} F ( x ) = Γ ( α ) γ ( α , λ x ) 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 ] = α λ E[X] = \dfrac{\alpha}{\lambda} E [ X ] = λ α Var(X) Var ( X ) = α λ 2 \operatorname{Var}(X) = \dfrac{\alpha}{\lambda^{2}} Var ( X ) = λ 2 α M(t) M ( t ) = ( λ λ − t ) α , t < λ M(t) = \left(\dfrac{\lambda}{\lambda - t}\right)^{\alpha}, \quad t < \lambda M ( t ) = ( λ − t λ ) α , t < λ Support − ∞ < x < ∞ -\infty < x < \infty − ∞ < x < ∞ f(x) f ( x ) = 1 σ 2 π exp ( − ( x − μ ) 2 2 σ 2 ) f(x) = \dfrac{1}{\sigma\sqrt{2\pi}}\exp\!\left(-\dfrac{(x-\mu)^{2}}{2\sigma^{2}}\right) f ( x ) = σ 2 π 1 exp ( − 2 σ 2 ( x − μ ) 2 ) F(x) F ( x ) = Φ ( x − μ σ ) F(x) = \Phi\!\left(\dfrac{x-\mu}{\sigma}\right) F ( x ) = Φ ( σ x − μ ) No elementary antiderivative - Φ is read from the standard normal table. E[X] E [ X ] = μ E[X] = \mu E [ X ] = μ Var(X) Var ( X ) = σ 2 \operatorname{Var}(X) = \sigma^{2} Var ( X ) = σ 2 M(t) M ( t ) = exp ( μ t + σ 2 t 2 2 ) M(t) = \exp\!\left(\mu t + \dfrac{\sigma^{2}t^{2}}{2}\right) M ( t ) = exp ( μ t + 2 σ 2 t 2 ) f(x) f ( x ) = 1 x σ 2 π exp ( − ( ln x − μ ) 2 2 σ 2 ) f(x) = \dfrac{1}{x\sigma\sqrt{2\pi}}\exp\!\left(-\dfrac{(\ln x-\mu)^{2}}{2\sigma^{2}}\right) f ( x ) = x σ 2 π 1 exp ( − 2 σ 2 ( ln x − μ ) 2 ) F(x) F ( x ) = Φ ( ln x − μ σ ) F(x) = \Phi\!\left(\dfrac{\ln x-\mu}{\sigma}\right) F ( x ) = Φ ( σ ln x − μ ) E[X] E [ X ] = e μ + σ 2 / 2 E[X] = e^{\mu + \sigma^{2}/2} E [ X ] = e μ + σ 2 /2 Var(X) Var ( X ) = ( e σ 2 − 1 ) e 2 μ + σ 2 \operatorname{Var}(X) = \left(e^{\sigma^{2}}-1\right)e^{2\mu+\sigma^{2}} Var ( X ) = ( e σ 2 − 1 ) e 2 μ + σ 2 Support 0 < x < 1 0 < x < 1 0 < x < 1 f(x) f ( x ) = Γ ( α + β ) Γ ( α ) Γ ( β ) x α − 1 ( 1 − x ) β − 1 f(x) = \dfrac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1} f ( x ) = Γ ( α ) Γ ( β ) Γ ( α + β ) x α − 1 ( 1 − x ) β − 1 F(x) F ( x ) = I x ( α , β ) F(x) = I_{x}(\alpha, \beta) 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 ] = α α + β E[X] = \dfrac{\alpha}{\alpha+\beta} E [ X ] = α + β α Var(X) Var ( X ) = α β ( α + β ) 2 ( α + β + 1 ) \operatorname{Var}(X) = \dfrac{\alpha\beta}{(\alpha+\beta)^{2}(\alpha+\beta+1)} Var ( X ) = ( α + β ) 2 ( α + β + 1 ) α β f(x) f ( x ) = x k / 2 − 1 e − x / 2 2 k / 2 Γ ( k / 2 ) f(x) = \dfrac{x^{k/2-1}e^{-x/2}}{2^{k/2}\Gamma(k/2)} f ( x ) = 2 k /2 Γ ( k /2 ) x k /2 − 1 e − x /2 F(x) F ( x ) = γ ( k / 2 , x / 2 ) Γ ( k / 2 ) F(x) = \dfrac{\gamma(k/2,\, x/2)}{\Gamma(k/2)} F ( x ) = Γ ( k /2 ) γ ( k /2 , x /2 ) Var(X) Var ( X ) = 2 k \operatorname{Var}(X) = 2k Var ( X ) = 2 k M(t) M ( t ) = ( 1 − 2 t ) − k / 2 , t < 1 2 M(t) = (1-2t)^{-k/2}, \quad t < \tfrac{1}{2} M ( t ) = ( 1 − 2 t ) − k /2 , t < 2 1 f(x) f ( x ) = α θ α ( x + θ ) α + 1 f(x) = \dfrac{\alpha\theta^{\alpha}}{(x+\theta)^{\alpha+1}} f ( x ) = ( x + θ ) α + 1 α θ α F(x) F ( x ) = 1 − ( θ x + θ ) α F(x) = 1 - \left(\dfrac{\theta}{x+\theta}\right)^{\alpha} F ( x ) = 1 − ( x + θ θ ) α E[X] E [ X ] = θ α − 1 , α > 1 E[X] = \dfrac{\theta}{\alpha - 1}, \quad \alpha > 1 E [ X ] = α − 1 θ , α > 1 Var(X) Var ( X ) = α θ 2 ( α − 1 ) 2 ( α − 2 ) , α > 2 \operatorname{Var}(X) = \dfrac{\alpha\theta^{2}}{(\alpha-1)^{2}(\alpha-2)}, \quad \alpha > 2 Var ( X ) = ( α − 1 ) 2 ( α − 2 ) α θ 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.