Mathematical statistics

Continuous Distributions

Distribution

Uniform

Normal

Exponential

Gamma

Chi-square

Beta

Probability Function

f (y) =

f (y) =

1
; θ ≤ y ≤ θ2 θ2 − θ1 1

1
1
(y − µ)2
√ exp −
2
2σ σ 2π
−∞ < y < +∞

f (y) =

1 y α−1 e−y/β ;
(α)β α
0<y<∞

f (y) =

f (y) =

f (y) =

1 −y/β e ; β>0 β 0<y<∞

(y)(v/2)−1 e−y/2
2v/2 (v/2) y2 > 0

;

(α + β) y α−1 (1 − y)β−1 ;
(α) (β)
0<y<1

MomentGenerating
Function

Mean

Variance

θ1 + θ2
2

(θ2 − θ1 )2
12

µ

σ2

β

β2

(1 − βt)−1

αβ

αβ 2

(1 − βt)−α

v

2v

(1−2t)−v/2

α α+β αβ
(α + β)2 (α + β + 1)

does not exist in closed form

etθ2 − etθ1 t (θ2 − θ1 )

exp µt +

t 2σ 2
2

Discrete Distributions

Distribution
Binomial

Probability Function p(y) =

n y p y (1 − p)n−y ;

Mean

Variance

MomentGenerating
Function

np

np(1 − p)

[ pet + (1 − p)]n

1 p 1− p

pet
1 − (1 − p)et

y = 0, 1, . . . , n

Geometric

p(y) = p(1 − p) y−1 ;

p

y = 1, 2, . . .

Hypergeometric

p(y) =

r y N−r n−y N n ;

nr
N

n

r
N

2

N −r
N

N −n
N −1

y = 0, 1, . . . , n if n ≤ r, y = 0, 1, . . . , r if n > r

Poisson

Negative binomial

λ y e−λ
;
y! y = 0, 1, 2, . . .

p(y) =

p(y) =

y−1 r−1 p r (1 − p) y−r ;

y = r, r + 1, . . .

λ

λ

exp[λ(et − 1)]

r p r(1 − p)

pet
1 − (1 − p)et

p

2

r

MATHEMATICAL STATISTICS WITH APPLICATIONS

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SEVENTH EDITION

Mathematical
Statistics with
Applications
Dennis D. Wackerly
University of Florida

William Mendenhall III
University of Florida, Emeritus

Richard L. Scheaffer
University of Florida, Emeritus

Australia • Brazil • Canada • Mexico • Singapore • Spain
United Kingdom • United States

Mathematical Statistics with Applications, Seventh Edition
Dennis D. Wackerly, William Mendenhall III, Richard L. Scheaffer
Statistics Editor: Carolyn Crockett
Assistant Editors: Beth Gershman, Catie Ronquillo
Editorial Assistant: Ashley Summers
Technology Project Manager: Jennifer Liang
Marketing Manager: Mandy Jellerichs
Marketing Assistant: Ashley Pickering
Marketing

References: Berger, J. O. 1985. Statistical Decision Theory and Bayesian Analysis, 2d ed. New York: Springer-Verlag. Box, G. E. P. 1980. “Sampling and Bayes’ Inference in Scientific Modeling and Robustness,” J Box, G. E. P., and G. C. Tiao. 1992. Bayesian Inference in Statistical Analysis. New York: Wiley Classics. Casella, G., and R. L. Berger. 2002. Statistical Inference, 2d ed. Pacific Grove, Calif.: Duxbury. Hogg, R. V., J. W. McKean, and A. T. Craig. 2005. Introduction to Mathematical Statistics, 6th ed Kepner, J., and D. Wackerly. 2002. “Observations on the Effect of the Prior Distribution on the Predictive Distribution in Bayesian Inferences,” Journal of Mood, A. M., F. A. Graybill, and D. Boes. 1974. Introduction to the Theory of Statistics, 3d ed Rice, J. A. 1995. Mathematical Statistics and Data Analysis, 2d ed. Belmont, Calif.: Duxbury.