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?
Define an event.
An event is a collection of possible outcomes.
What symbol represents the impossible event?
What symbol represents a certain event?
Define a field.
Are fields open or closed w.r.t. taking finite unions or intersections?
Define a σ-field.
What can you replace property 2 by in the following?
What is the smallest σ-field in 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?
What is the pair (Ω,𝑭) called?
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 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.?