Theory of Statistics - George Mason University

676 0 Statistical Mathematics
Multivariate integrals
Both the normal distribution and the beta distribution have important
and straightforward multivariate extensions. These are associated with
important multivariate integrals.
• over IR d ; Aitken’s integral:
e −(x−µ)TΣ −1 (x−µ)/2 d/2 1/2
dx = (2π) |Σ| , (0.0.88)
IR d
for positive definite Σ −1 .
• over ]0, 1[ d ; Dirichlet integral:
d
]0,1[ d
x
i=1
αi−1
i
1 −
d
i=1
xi
αd+1−1
dx =
d+1 i=1 Γ(αi)
Γ( . (0.0.89)
d+1
i=1 αi)
Series
There are four simple series that should also be immediately recognizable:
• over 0, . . ., n; the binomial series:
n
x=0
Γ(n + 1)
Γ(x + 1)Γ(n − x + 1) πx (1 − π) n−x = 1, (0.0.90)
for 0 < π < 1 and n ≥ 1.
• over max(0, N − L + M), . . ., min(N, M); the hypergeometric series:
min(N,M)
x=max(0,N−L+M)
M
x
L − M
=
N − x
for 1 ≤ L, 0 ≤ N ≤ L, and 0 ≤ M ≤ L.
• over 0, 1, 2, . . .; the geometric series:
∞
x=0
for 0 < π < 1.
• over 0, 1, 2, . . .; the Poisson series:
for θ > 0.
(1 − π) x = π −1
∞
x=0
L
, (0.0.91)
n
(0.0.92)
θ x
x! = eθ , (0.0.93)
The beta integral and the binomial series have a natural connection
through the relation
Γ(n + 1)
Γ(x + 1)Γ(n − x + 1) =
n
. (0.0.94)
x
Theory of Statistics c○2000–2013 James E. Gentle