Page 53 - Probability and Statistical Inference

30 1. Notions of Probability

Differentiation Under Integration: Suppose that f(x, θ) is a differen-
tiable function in θ, for x ∈ ℜ, θ ∈ ℜ. Let there be another function g(x, θ)
such that (i) |∂f(x, θ)/∂θ| | ≤ g(x, θ) for all θ belonging to some interval
θ=θ0 0
(θ – ε, θ + ε), and (ii) . Then

Monotone Function of a Single Real Variable: Suppose that f(x) is a
real valued function of x ∈ (a, b) ⊆ ℜ. Let us assume that d f(x)/dx exists at
each x ∈ (a, b) and that f(x) is continuous at x = a, b. Then,

Gamma Function and Gamma Integral: The expression Γ(α), known
as the gamma function evaluated at α, is defined as

The representation given in the rhs of (1.6.19) is referred to as the gamma
integral. The gamma function has many interesting properties including the
following:

Stirling’s Approximation: From equation (1.6.19) recall that Γ(α) =
with α > 0. Then,

Writing α = n + 1 where n is a positive integer, one can immediately claim
that

The approximation for n! given by (1.6.22) works particularly well even for n
as small as five or six. The derivation of (1.6.22) from (1.6.21) is left as the
Exercise 1.6.15.