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Flashcards in Preliminaries Deck (65)
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1

Define a sample space.

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

2

What symbol represents a sample space?

Ω

3

Define an event.

An event is a collection of possible outcomes.

4

What symbol represents the impossible event?

5

What symbol represents a certain event?

Ω

6

Define a field.

7

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

Closed

8

Define a σ-field.

9

What can you replace property 2 by in the following?

10

What is the smallest σ-field in Ω?

{∅,Ω}

11

What is the biggest σ-field in Ω?

All the subsets of Ω.

12

Define a probability distribution.

13

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

14

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

15

Define a probability space.

16

What is another name for a probabiluty space?

Probability measure.

17

What is the pair (Ω,𝑭) called?

Measurable space

18

Finish the following lemma.

19

Prove the following Lemma

Need to take photo.

20

Define conditional probability.

21

What is P4 - the multiplication rule for probabilities?

22

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

23

What is P6 - Bayes' theorem?

24

Define independent.

25

Define mutually independent.

26

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.

27

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

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

28

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.

29

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

30

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