Theory of Statistics - George Mason University

858 Appendix C. Notation and Definitions
P(α, x) The regularized incomplete gamma function, which is
the CDF of the standard gamma distribution,
B(α, β) The beta function,
P(α, x) =
B(α, β) = Γ(α)Γ(β)
Γ(α + β)
=
=
γ(α, x)
. (C.6)
Γ(α)
(C.7)
1
t
0
α−1 (1 − t) β−1 dt (C.8)
∞
t
0
α−1
.
(1 + t) α+β
(C.9)
The integral in equation (C.8) is called Euler’s beta integral.
Ix(α, β) The regularized incomplete beta function, which is the
CDF of the beta distribution,
Ix(α, β) =
1
B(α, β)
C.3 Sets, Measure, and Probability
x
t
0
α−1 (1 − t) β−1 dt. (C.10)
The notation listed below does not always represent the things associated
with it here, but for these objects, I generally use either this notation or other
symbols in the same font.
Ω Sample space; the universal set in a given space or probability
distribution.
#A The cardinality of the set A.
A c The complement of the set A; A c = Ω − A.
A1 ∪ A2
The union of the sets A1 and A2; x ∈ A1 ∪A2 iff x ∈ A1
or x ∈ A2.
Theory of Statistics c○2000–2013 James E. Gentle