Preliminaries

Question 1

Define a sample space.

Answer 1

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

Question 2

What symbol represents a sample space?

Answer 2

Ω

Question 3

Define an event.

Answer 3

An event is a collection of possible outcomes.

Question 4

What symbol represents the impossible event?

Answer 4

∅

Question 5

What symbol represents a certain event?

Answer 5

Ω

Question 6

Define a field.

Answer 6

Question 7

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

Answer 7

Closed

Question 8

Define a σ-field.

Answer 8

Question 9

What can you replace property 2 by in the following?

Answer 9

Question 10

What is the smallest σ-field in Ω?

Answer 10

{∅,Ω}

Question 11

What is the biggest σ-field in Ω?

Answer 11

All the subsets of Ω.

Question 12

Define a probability distribution.

Answer 12

Question 13

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

Answer 13

Question 14

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

Answer 14

Question 15

Define a probability space.

Answer 15

Question 16

What is another name for a probabiluty space?

Answer 16

Probability measure.

Question 17

What is the pair (Ω,𝑭) called?

Answer 17

Measurable space

Question 18

Finish the following lemma.

Answer 18

Question 19

Prove the following Lemma

Answer 19

Need to take photo.

Question 20

Define conditional probability.

Answer 20

Question 21

What is P4 - the multiplication rule for probabilities?

Answer 21

Question 22

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

Answer 22

Question 23

What is P6 - Bayes’ theorem?

Answer 23

Question 24

Define independent.

Answer 24

Question 25

Define mutually independent.

Answer 25

Question 26

Define a random variable, X.

Answer 26

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.

Question 27

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

Answer 27

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

Question 28

When is X a discrete r.v.?

Answer 28

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

Question 29

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

Answer 29

Question 30

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

Answer 30

Question 31

What is the probability distribution for a continuous r.v. X?

Answer 31

Question 32

Define the cumulative distribution function.

Answer 32

Question 33

Define a multivariate random variable.

Answer 33

Question 34

What is the joint probability distribution of (X,Y)?

Answer 34

Question 35

What is the marginal probability distribution of X in a joint distribution?

Answer 35

Question 36

What is the probability mass function for the conditional distribution of X given Y?

Answer 36

Question 37

What is the r.v. version of the partion theorem called?

Answer 37

Law of total probability.

Question 38

What is the law of total probability?

Answer 38

Question 39

If X and Y are independent what does p(x,y) factorise to?

Answer 39

Question 40

Define when two random variable X, Y are independent.

Answer 40

Question 41

What is the continuous probability density function for a joint distribution of (X,Y)?

Answer 41

Question 42

What are the marginal pdfs for the joint, continuous r.v. (X,Y)?

Answer 42

Question 43

What is the continuous conditional density distribution of X given Y?

Answer 43

Question 44

Define the expected value.

Answer 44

Question 45

What are the names of the four important properties of expectation?

Answer 45

1. Linearity
2. Monotonicity
3. Multivariate linearity
4. Independence

Question 46

What is the equation for variance?

Answer 46

Question 47

Define conditional expectation.

Answer 47

Question 48

What is the E1 - linearity - property of expectation?

Answer 48

Question 49

What is the formula for covariance?

Answer 49

Question 50

What does covariance equal to show two varaibles are uncorrelated?

Answer 50

0

Question 51

What does the following equal?

Answer 51

Question 52

What does the following equal when the variables are pairwise uncorrelated?

Answer 52

Question 53

What is the partition theorm for expectation?

Answer 53

Question 54

What is the law of large numbers theorem?

Answer 54

Question 55

What is the central limit theorem?

Answer 55

Question 56

Define the moment generating function.

Answer 56

Question 57

What are the name of the five useful properties of the moment generating function?

Answer 57

1. Expectation
2. Uniqueness
3. Linear transformation
4. Independence
5. Convergence

Question 58

What is the M1 - expectation - property of moment generating functions?

Answer 58

Question 59

What is the M2 - uniqueness - property of moment generating functions?

Answer 59

Question 60

What is the M3 - linear transformation - property of moment generating functions?

Answer 60

Question 61

What is the M4 - independence - property of moment generating functions?

Answer 61

Question 62

What is the M5 - convergence - property of moment generating functions?

Answer 62

Question 63

What is the E2 - monotonicity - property of expectation?

Answer 63

Question 64

What is the E3 - multivaraite independence - property of expectation?

Answer 64

Question 65

What is the E4 - independence - property of expectation?

Answer 65