Independence

Question 1

When are events independent

Answer 1

Events are indep when P(A N B ) = P(A) x P(B)
P(A|B) = P(A) = P(A|B^C)

Question 2

What is P(of finitely independent events)

Answer 2

P(finitely independent events) = PI P(events)

Question 3

When are infinitely many events independent

Answer 3

Infinitely many events are independent when for all n in N, E1,…,En are independent

Question 4

What is the 2nd Borel-Cantelli lemma

Answer 4

2nd Borel -Cantelli is:
If An is a collection of independent events s.t sum k>=1 P(Ak) = inf then events An happen infinitely many times w.p 1

Question 5

When is a finite or infinite sequence X1, X2… of RVs independent

Answer 5

A finite or infinite sequence X1, X2,… of RVs independent when (X1 <= x1), (X2 <= x2) re independent

Question 6

What is equivalent definition for a sequence of RVs to be independent

Answer 6

Equivalent definition for a sequence of RVs to be independent is:
P(p=1 to k N Xip <= xip) = PI p=1 to k P(Xip <= xip)

Question 7

If X1,X2 … are independent and f1,f2 …:R to R, then what are f1(X1),…

Answer 7

F1(X1)… are independent RVs

Question 8

What is E[XY] if X,Y are independent RVs

Answer 8

E[XY] = E[X] E[Y] if X,Y are independent RVs

Question 9

If X and Y E L1 (integrable), then what is E[XY]

Answer 9

If X and Y E L1 then XY E L1 and E[XY] = E[X]E[Y]