Page 52 - Probability and Statistical Inference

1. Notions of Probability 29

Results on Limits:

The Big O(.) and o(.) Terms: Consider two terms a and b , both are real
n
n
valued but depend on n = 1, 2, ... . The term a is called O(b ) provided that a /
n
n
n
b → c, a constant, as n → ∞. The term a is called o(b ) provided that a /b →
n
n
n
n
n
0 as n → ∞. Let a = n – 1, b =2n+ n and d = 5n + n, n = 1, 2, ... . Then,
8/7
n
n
n
one has, for example, a = O(b ), a = o(d ), b = o(d ). Also, one may write,
n
n
n
n
n
n
for example, a = O(n), b = O(n), d = O(n ), d = o(n ), d = o(n ).
2
8/7
9/7
n n n n n
Taylor Expansion: Let f(.) be a real valued function having the finite n th
derivative d f(x)/dx , denoted by f (.), everywhere in an open interval (a, b)
n
n
(n)
⊆ ℜand assume that f (n–1 )(.) is continuous on the closed interval [a, b]. Let c
∈ [a, b]. Then, for every x ∈ [a, b], x ≠ c, there exists a real number ξ
between the two numbers x and c such that
Some Infinite Series Expansions:
Differentiation Under Integration (Leibnitz’s Rule): Suppose that
f(x, θ), a(θ), and b(θ) are differentiable functions with respect to θ for x ∈
ℜ, θ ∈ ℜ. Then,