Lecture 1

Why we study probability and random processes

• The world around us is random
• Comms systems are impaired by random noise
• Computer network traffic patterns are random
• clocks have jitter
• cant control randomness, but can understand and use that understanding to our advantage

Basic Set Theory

• A set $S$ is a collection of objects which are elements of the set
• If $x$ is an element of $S$, we say $x \in S$
• If $x$ is not an element of $S$, we say $x \in S$
• If S has no elements, it is called the empty set $\emptyset$
• Sets can be:
  • Finite:
    • ex: $\{ 1, 2, 3, 4 \}$
  • Countable infinite
    • ex: $\mathbb{Z}$
  • Uncountable
    • ex: $\mathbb{R}$

Relationships

Subsets

$$S \in T$$

Equality

If every element of the set $S$ is also an element of set $T$, we say $S$ is a "subset of $T$"

If $S \subset T$ and $T \subset S$, then $S = T$.

Universal Set $\Omega$

The set of all objects in the particular context.

Complement

The set of all elements of $\Omega$ that do not belong to $S$

can be written as $S^{C}$ or $S^\prime$

Union

The set of all elements that belong to $S$ or $T$

Intersection

The set of all elements that belong to both $S$ and $T$

Algebraic Properties

Commutative

$$ S \cup T = T \cup S $$

$$ S \cap T = T \cap S $$

Associative

$$ S \cup (T \cup W) = (S \cup T) \cup W $$

$$ S \cap (T \cap W) = (S \cap T) \cap W $$

Distributive

$$ S\cup \left( T\cap W \right) =\left( S\cup T \right) \cap \left( S\cup W \right) $$

$$ S\cap \left( T\cap W \right) =(S\cap T)\cup(S\cap W) $$

De Morgan's Laws

$$ (S\cap T)^\prime=S^\prime \cup T^\prime $$

$$ (S\cup T)^\prime=S^\prime\cap T^\prime $$