Reference tables for common probability distributions.
Note: In practice, R computes these values much more precisely than any printed table. These tables are provided for exam reference and hand calculations.
1 Standard Normal Distribution Table ¶ The table below gives P ( Z ≤ z ) = Φ ( z ) P(Z \leq z) = \Phi(z) P ( Z ≤ z ) = Φ ( z ) Z ∼ N ( 0 , 1 ) Z \sim N(0, 1) Z ∼ N ( 0 , 1 )
This table will be generated programmatically when the chapter on the Normal distribution (Chapter 5) is completed.
2 Common Distribution Summary ¶ Distribution PMF/PDF E(X) V(X) MGF Binomial( n , p ) (n, p) ( n , p ) ( n x ) p x ( 1 − p ) n − x \binom{n}{x}p^x(1-p)^{n-x} ( x n ) p x ( 1 − p ) n − x n p np n p n p ( 1 − p ) np(1-p) n p ( 1 − p ) ( p e t + 1 − p ) n (pe^t + 1-p)^n ( p e t + 1 − p ) n Geometric( p ) (p) ( p ) p ( 1 − p ) x − 1 p(1-p)^{x-1} p ( 1 − p ) x − 1 1 / p 1/p 1/ p ( 1 − p ) / p 2 (1-p)/p^2 ( 1 − p ) / p 2 p e t / [ 1 − ( 1 − p ) e t ] pe^t/[1-(1-p)e^t] p e t / [ 1 − ( 1 − p ) e t ] Poisson( λ ) (\lambda) ( λ ) e − λ λ x / x ! e^{-\lambda}\lambda^x/x! e − λ λ x / x ! λ \lambda λ λ \lambda λ e λ ( e t − 1 ) e^{\lambda(e^t - 1)} e λ ( e t − 1 ) Uniform( a , b ) (a,b) ( a , b ) 1 / ( b − a ) 1/(b-a) 1/ ( b − a ) ( a + b ) / 2 (a+b)/2 ( a + b ) /2 ( b − a ) 2 / 12 (b-a)^2/12 ( b − a ) 2 /12 ( e t b − e t a ) / [ t ( b − a ) ] (e^{tb}-e^{ta})/[t(b-a)] ( e t b − e t a ) / [ t ( b − a )] Normal( μ , σ 2 ) (\mu, \sigma^2) ( μ , σ 2 ) 1 σ 2 π e − ( x − μ ) 2 / ( 2 σ 2 ) \frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/(2\sigma^2)} σ 2 π 1 e − ( x − μ ) 2 / ( 2 σ 2 ) μ \mu μ σ 2 \sigma^2 σ 2 e μ t + σ 2 t 2 / 2 e^{\mu t + \sigma^2 t^2/2} e μ t + σ 2 t 2 /2 Exponential( β ) (\beta) ( β ) 1 β e − x / β \frac{1}{\beta}e^{-x/\beta} β 1 e − x / β β \beta β β 2 \beta^2 β 2 ( 1 − β t ) − 1 (1-\beta t)^{-1} ( 1 − βt ) − 1 Gamma( α , β ) (\alpha, \beta) ( α , β ) x α − 1 e − x / β β α Γ ( α ) \frac{x^{\alpha-1}e^{-x/\beta}}{\beta^\alpha\Gamma(\alpha)} β α Γ ( α ) x α − 1 e − x / β α β \alpha\beta α β α β 2 \alpha\beta^2 α β 2 ( 1 − β t ) − α (1-\beta t)^{-\alpha} ( 1 − βt ) − α Beta( α , β ) (\alpha, \beta) ( α , β ) x α − 1 ( 1 − x ) β − 1 B ( α , β ) \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)} B ( α , β ) x α − 1 ( 1 − x ) β − 1 α α + β \frac{\alpha}{\alpha+\beta} α + β α α β ( α + β ) 2 ( α + β + 1 ) \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)} ( α + β ) 2 ( α + β + 1 ) α β complex