4) Independence

Question 1

What does it mean for events {Ai } i∈I
to be independent

Answer 1

Question 2

What does it mean for families of events {Gi} i∈I to be independent

Answer 2

Question 3

What does it mean for random variables {Xi}i∈I to be independent

Answer 3

Question 4

What does the π-system property tell us about independence in probability spaces

Answer 4

Question 5

What condition characterises the independence of random variables {Xi}i∈I

Answer 5

Question 6

What are the key notations used for a function F:Rn
→R and sequences in R

Answer 6

Question 7

What are the conditions for a function F:R^n →[0,1] to be a distribution function

Answer 7

Question 8

What does Kolmogorov’s Consistency Theorem state

Answer 8

Question 9

What guarantees the existence of a sequence of i.i.d. random variables with a given distribution function F

Answer 9

Question 10

What does the Second Borel-Cantelli Lemma state

Answer 10

Question 11

What is the tail σ-algebra of a sequence of random variables

Answer 11

Question 12

What are some key events and random variables in the tail σ-algebra of a sequence X=(Xn) n≥1

Answer 12

Question 13

What properties does the tail σ-algebra TX of a sequence of independent random variables X=(Xn) n≥1 satisfy

Answer 13