title: Some Random Series Functions 2ed (Cambridge Studies in Advanced Mathematics)
author: Kahane
description: "...a rewarding [book], certain to have a wide influence." Robert Kaufman, American Scientist
zlib_text: Moment-
  Generating
  Distribution
  Probability Function
  Mean
  Variance
  Function
  Uniform
  f (y) =
  1
  θ2 − θ1
  ; θ1 ≤ y ≤ θ2
  θ1 + θ2
  2
  (θ2 − θ1)2
  12
  etθ2 − etθ1
  t(θ2 − θ1)
  Normal
  f (y) =
  1
  σ
  √
  2π
  exp
  �
  −
  � 1
  2σ 2
  �
  (y − µ)2
  �
  µ
  σ 2
  + t2σ 2
  2
  �
  −∞ < y < +∞
  Exponential
  f (y) = 1
  β e−y/β;
  β > 0
  β
  β2
  (1 − βt)−1
  0 < y < ∞
  Gamma
  f (y) =
  �
  1
  �(α)βα
  �
  yα−1e−y/β;
  αβ
  αβ2
  (1 − βt)−α
  0 < y < ∞
  Chi-square
  f (y) = (y)(v/2)−1e−y/2
  2v/2�(v/2)
  ;
  v
  2v
  (1−2t)−v/2
  y2 > 0
  Beta
  f (y) =
  � �(α + β)
  �(α)�(β)
  �
  yα−1(1 − y)β−1;
  α
  α + β
  αβ
  (α + β)2(α + β + 1)
  does not exist in
  closed form
  0 < y < 1
  Continuous Distributions
  exp  µt
  
  Moment-
  Generating
  Distribution
  Probability Function
  Mean
  Variance
  Function
  Binomial
  p(y) =
  �
  n
  y
  �
  py(1 − p)n−y;
  np
  np(1 − p)
  [pet + (1 − p)]n
  y = 0, 1, . . . , n
  Geometric
  p(y) = p(1 − p)y−1;
  1
  p
  1 − p
  p2
  pet
  1 − (1 − p)et
  y = 1, 2, . . .
  Hypergeometric
  p(y) =
  �
  r
  y
  � �
  N−r
  n−y
  �
  �
  N
  n
  �
  ;
  nr
  N
  n
  � r
  N
  � � N − r
  N
  � � N − n
  N − 1
  �
  y = 0, 1, . . . , n if n ≤ r,
  y = 0, 1, . . . , r if n > r
  Po