Probability Spaces

Question 1

What is sample space

Answer 1

Sample space is set of all possible outcomes (omega)

Question 2

What is F symbol (if omega is countably infinite or finite)

Answer 2

F symbol is the event space, set of all subsets of omega or the power set of omega

Question 3

What is indicator of F and empty set

Answer 3

Indicator of F is 1f (w) and 1empty set(w) = 0

Question 4

What is relation between indicator of F and F^C

Answer 4

Relationship between indicator of F and F^C is:
1f^c (w) = 1 - 1f(w)

Question 5

What is indicator of all Fn s occurring (N n=1 to inf fn)

Answer 5

Indicator of all Fn s occurring is :
PI n=1 to inf 1fn(w)

Question 6

What is indicator at least one fn happens

Answer 6

Indicator at least one fn happens is:
(U n=1 to inf fn)^C = N n=1 to inf (fn)^C

Question 7

What is an example of power set of omega

Answer 7

Example of power set of omega is:
Coin flip once, omega = (H,T)
P(omega) = H, T , (H,T), empty set

Question 8

What is sigma algebra/field

Answer 8

Sigma algebra/field is:
A subset F of P(omega) is a sigma algebra if
Empty set E F
If subs F E F then F^C E (stable under complement)
If Fn E F for all n >=1, then Un=1 to inf Fn E F (stable und countable Union)
F is not empty

Question 9

What are implication of sigma algebra

Answer 9

Implications of sigma algebra are:
Omega E F
N n=1 to inf Fn E F (stable under countable intersection)

Question 10

What is an example of a sigma field

Answer 10

An example of a sigma field is:
F = (empty set, omega) (something happens or nothing happens)
Take E in omega
F = (empty set, omega, E, E^C)

Question 11

If omega is countable, what is P(omega)

Answer 11

If omega is countable, then P(omega) is a sigma field