Define a **sample space**.

A **sample space** Ω is a collection of all possible outcomes of a probabilistic experiment.

What symbol represents a sample space?

Ω

Define an **event.**

An **event** is a collection of possible outcomes.

What symbol represents the impossible event?

∅

What symbol represents a certain event?

Ω

Define a **field.**

Are fields open or closed w.r.t. taking finite unions or intersections?

Closed

Define a **σ-field.**

What can you replace property 2 by in the following?

What is the smallest σ-field in Ω?

{∅,Ω}

What is the biggest σ-field in Ω?

All the subsets of Ω.

Define a **probability distribution.**

What are the A1-A4 properties of a probability distribution?

What are the P1-P3 properties that following on from the following properties?

Define a **probability space.**

What is another name for a probabiluty space?

Probability measure.

What is the pair (Ω,𝑭) called?

Measurable space

Finish the following lemma.

Prove the following Lemma

Need to take photo.

Define **conditional probability.**

What is P4 - the multiplication rule for probabilities?

What is P5 - partition theorem or formula of total probabilities?

What is P6 - Bayes' theorem?

Define **independent.**

Define **mutually independent**.

Define a **random variable, X**.

If the sample space of possible outcomes is a set of real outcomes, then the outcome to the probabilistic experiment is called a **random variable.**

What is the probability distribution of a r.v. X?

The collection of probabilites ℙ(X ∈ A) for all intervals A ⊆ ℝ.

When is X a discrete r.v.?

If in addition Ω is countable, i.e. if the possible values for X can be enumerated in a (possiby infinite) list.

What is the probability mass function of a discrete r.v. X?

What is the probability distribution when X is a discrete r.v.?