Page 54 - Probability and Statistical Inference

1. Notions of Probability 31

Ratios of Gamma Functions and Other Approximations: Recall that
. Then , one has

with fixed numbers c(> 0) and d, assuming that the gamma function itself is
defined. Also, one has

with fixed numbers c and d, assuming that the gamma functions involved are
defined.
Other interesting and sharper approximations for expressions involving the
gamma functions and factorials can be found in the Section 6.1 in Abramowitz
and Stegun (1972).
Beta Function and Beta Integral: The expression b(α, β), known as the
beta function evaluated at α and β in that order, is defined as

The representation given in the rhs of (1.6.25) is referred to as the beta inte-
gral. Recall the expression of Γ(α) from (1.6.19). We mention that b(α, β) can
alternatively be expressed as follows:

Maximum and Minimum of a Function of a Single Real Variable: For
some integer n ≥ 1, suppose that f(x) is a real valued function of a single real
variable x ∈ (a, b) ⊆ R, having a continuous n derivative , denoted
th
(n)
by f (x), everywhere in the open interval (a, b). Suppose also that
(2)
(1)
for some point ξ ∈ (a, b), one has f (ξ) = f (ξ) = ... = f (n–
1) (ξ) = 0, but f (ξ) ≠ 0. Then,
(n)

Maximum and Minimum of a Function of Two Real Variables:
Suppose that f(x) is a real valued function of a two-dimensional variable