STAT 3923/4023
Semester 2 Statistical Inference Advanced 2021
Week 1
For general random variables we need to develop a theory of probability that will allow for
non-countably infinite sample spaces. Probability Theory was given a formal foundation
within Measure Theory by A. N. Kolmogorov (1933). Given a sample space Ω we restrict
attention to events in a σ-field F . F is a class of sets (or events) that satisfy
(i) φ ∈ F
(ii) A ∈ F ⇒ Ac ∈ F
(iii) A1, A2, .. ∈ F ⇒ ∪∞i=1 Ai ∈ F .
F is closed under complementation and finite and countable unions. Since (A∪B)c = Ac∩Bc
it is also closed under countable intersections.
The revised axioms of probability are
1. P (E) ≥ 0, E ∈ F
2. P (Ω) = 1
3. If E1, E2, .. is a countable sequence of disjoint events then
P (∪∞i=1Ei) =
∞∑
i=1
P (Ei).
(Ω,F , P ) is called a Probability Space.
Probability Properties
(i) A ⊂ B ⇒ P (A) ≤ P (B).
(ii) P (A ∪B) = P (A) + P (B)− P (A ∩B).
(iii) Boole’s inequality: P (∪ni=1Ai) ≤
∑n
i=1 P (Ai).
Random Variables
When dealing with the real line, the σ-field generated by the intervals (a, b], a ≤ b ∈ IR, is
called the Borel σ-field, B. B consists of half-open intervals, their complements, and the sets
obtained via countable unions and intersections of such intervals. A random variable is a
function from Ω to IR such that for all A ∈ B, X−1(A) ∈ F where
X−1(A) = {ω ∈ Ω : X(ω) ∈ A}.
X is a measurable function. The distribution function associated with X is
F (x) = P (X(ω) ≤ x) = P ({ω : X(ω) ∈ (−∞, x]}).
Properties of F .
(i) 0 ≤ F (x) ≤ 1.
(ii) If x < y then F (x) ≤ F (y), so F is non-decreasing.
(iii) F (−∞) = 0 and F (∞) = 1.
(iv) P (x < X ≤ y) = F (y)− F (x).
(v) F (x) is continuous to the right, limh→0 F (x+ h) = F (x), h > 0.
Let {xn} be a sequence increasing to x. Then P (xn < X ≤ x) = F (x)− F (xn) so
F (x)− lim
n→∞F (xn) = F (x)− F (x− 0) = P (X = x) ≥ 0.
Distribution functions can have jumps. The number of discontinuities is at most countably
infinite.
Examples
1. Let X be a random variable (rv) with distribution function
F (x) =
1
2(1 + x2)
if x ≤ 0, and F (x) = 1 + 2x
2
2(1 + x2)
if x > 0,
show that X is a continuous random variable and find its density.
F is piecewise continuous, F (0) = 1
2
, F (∞) = 1, F (−∞) = 0. As x tends to 0 from the
right F (x)→ 1
2
so F is continuous. Check that F has density
f(x) =
|x|
(1 + x2)2
.
2. Find the distribution function of the so-called ’extreme-value’ density
f(x) = exp(−x− exp(−x)), x ∈ IR.
F (x) =
∫ x
−∞
f(u) du
=
∫ x
−∞
e−ue−e
−u
du, set v = e−u,
=
∫ e−x
∞
−e−v dv
= exp[−e−x], x ∈ IR.

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