Probability and Stochastic Processes

PROBABILITY AND STOCHASTIC PROCESS (MTL106)

ANANTA KUMAR MAJEE




1. Probability Theory:
Consider the experiment of tossing a coin. There are two possible outcome {H, T } and we
don’t know whether H or T will come in advance for any performance of this experiment. From
this example, we arrive at the following:
Definition 1.1 (Random Experiment). A random experiment is an experiment in which
i) all outcomes of the experiment are known in advance
ii) any performance of the experiment results in an outcome that is not known in advance
iii) the experiment can be repeated under identical conditions.
Definition 1.2 (Sample space). The set Ω of all possible outcomes of a random experiment
is called sample space. An element ω ∈ Ω is called a sample point.
A die is thrown once. We are interested in the possible occurrences of the ’events’
a) the outcome is the number 1
b) the outcome is even number
c) the outcome is even but does not exceed 3
d) the outcome is not even
Each of the above events can be specified as a subset A of the sample space Ω = {1, 2, 3, 4, 5, 6}
as follows:
a) A = {1}, b) A = {2, 4, 6}, c) A = {2, 4, 6} ∩ {1, 2, 3}, d) A = {2, 4, 6}∁
Thus, events are subsets of Ω, but need all the subsets of Ω be events? The answer is NO. We
need the collection of events be closed under complement and countable union.
Definition 1.3 (σ-algebra). Let Ω ̸= ∅. A collection F of subsets of Ω is called a σ-algebra
or σ-field over Ω if the following properties hold:
i) ∅ ∈ F
ii) If A1 , A2 , . . . ∈ F, then ∪∞
i=1 Ai ∈ F
iii) If A ∈ F, then A . ∁


The elements of F are called ‘events’. One can easily show that if A, B ∈ F, then A ∩ B ∈ F.
Example 1.1. i) The smallest σ-field associated to Ω is the collection F = {∅, Ω}.
ii) If A is any subset of Ω, then F = {∅, A, A∁ , Ω} is a σ-field.
ii) P(Ω) := {A : A ⊆ Ω} is a σ-field, called the total σ-field over Ω.
Remark 1.1. In general, union of σ-algebras over Ω may not be a σ-algebra. For example, let
Ω = {1, 2, 3} and F1 = {∅, {1}, {2, 3}, Ω} and F2 = {∅, {1, 2}, {3}, Ω}. Then both F1 and F2 are
σ-algebra, but F1 ∪ F2 is NOT a σ-algebra.
Lemma 1.1. Intersection of two σ-fields over Ω is a σ-field over Ω. More generally, if {Fi :
i ∈ I} is a family of σ-fields over Ω, then G = ∩i∈I Fi is also a σ-field.
1

, 2 A. K. MAJEE

Definition 1.4 (Generated σ-field). Let Ω ̸= ∅ and A be be collection of subsets of Ω. Let
M := {F : F is a σ-field over Ω containing A}.
Then σ(A) := ∩F ∈M F is the smallest σ-field over Ω containing A. σ(A) is called the σ-field
generated by A.
Example 1.2 (Borel σ-field). The smallest σ-algebra over R containing all intervals of the
form (−∞, a] with a ∈ R is called the Borel σ-algebra and denoted by B(R).
• Any A ∈ B(R) is called a Borel subset of R.
• The followings are Borel subsets of R: for a ≤ b
1
(a, ∞) = R \ (−∞, a], (a, b] = (−∞, b] ∩ (a, ∞), (∞, a) = ∪∞ n=1 (−∞, a − ],
 n 
[a, ∞) = R \ (−∞, a), (a, b) = (−∞, b) ∩ (a, ∞), [a, b] = R \ (−∞, a) ∪ (b, ∞) ,
m
{a} = [a, a], N = ∪∞ n=0 {n}, Q = ∪m,n∈Z,n̸=0 { }, Q∁ = R \ Q.
n
Example 1.3 (Borel n
Q σ-algebra on R ). It is σ-algebra generated by the n-dimensional rect-
angles of the form ni=1 (ai , bi ]
Definition 1.5 (Measurable space). Let Ω ̸= ∅ and F be a sigma-algebra over Ω. The pair
(Ω, F) is called a measurable space.
Our goal now is to assign to each event A, a non-negative real number which indicates its
chance of happening.
Definition 1.6 (Probability measure). Let (Ω, F) is a measurable space. A function P :
F → [0, 1] is called probability measure if
i) P(A) ≥ 0 ∀ A ∈ F
ii) P(∅) = 0 and P(Ω) = 1
iii) If {Ai } ⊂ F and Ai ’s are pairwise disjoint, then
∞

X
P ∪∞
i=1 A i = P(Ai ).
i=1
The triplet (Ω, F, P) is called a probability space.
Example 1.4. Let Ω = {1, 2, 3} and F = {∅, Ω, {1}, {2, 3}}. Define P : F → [0, 1] via
(
1, if 3 ∈ A
P(A) =
0, if 3 ∈ / A.
Then P is a probability measure on (Ω, F).
Example 1.5. Let Ω = (0, ∞) and F is the Borel σ-field on Ω. Define, for each interval A ⊂ Ω
Z
P(A) = e−x dx.
A
R −x
Observe that P(A) ≥ 0 for all A ∈ F and Ω e dx = 1. Let {Ai } ⊂ B and pairwise disjoint.
∞

R −x
Then P ∪i=1 Ai = R e 1∪∞ i=1 Ai
(x) dx. We need to show that
Z ∞ Z
X ∞
X
−x −x
e 1∪∞ i=1 Ai
(x) dx = e dx = P(Ai ).
R i=1 Ai i=1
One can show the first equality by using monotone convergence theorem on the sequence of
functions fn (x) = e−x 1∪ni=1 Ai (x).