Probability Theory

Question 1

What is the first axiom of probability measures?

Answer 1

P(W) = 1 and P(Ø) = 0

Question 2

What is the second axiom of probability measures?

Answer 2

If A ∩ B = ø then P(A ∪ B) = P(A) + P(B)

Question 3

Define the property of General Additivity

Answer 3

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Question 4

Denote the uncertainty measure

Answer 4

µ = P

Question 5

Define the complement

Answer 5

P(A^∁) = 1 - P(A)

Question 6

How many values does an agent need to specify to define an uncertainty measure?

Answer 6

2^n - 2

Question 7

How many values does an agent need to specify to define a probability distribution?

Answer 7

n - 1

Question 8

For A, B ⊆ W, how is the conditional probability of A given B defined?

Answer 8

P(A|B) = P(A ∩ B) / P(B)

Question 9

Define Bayes Theorem

Answer 9

P(A|B) = P(B|A)P(A) / P(B)

Question 10

Define the theorem of total probaility

Answer 10

P(A) = P(A|B)P(B) + P(A|B^c)P(B^c)

Question 11

What is a prior in the context of probability theory?

Answer 11

A defined probability measure

Question 12

What is Laplace’s principle of insufficient reason?

Answer 12

In the absence of any other information all possible worlds should be assumed to be equally probable

Question 13

What two justifications are there for choosing probability to measure uncertainty?

Answer 13

Cox’s Justification and Willingness to bet (de Finetti, Ramsay, Kemeny)

Question 14

Define Cox’s first axiom

Answer 14

The agent defines a conditional measure

Question 15

Define Cox’s second axiom

Answer 15

If A ≠ Ø then

Question 16

Define Cox’s third axiom

Answer 16

If B ≠ Ø

then

Question 17

Define Cox’s fourth axiom

Answer 17

If B∩C ≠ Ø

then

Question 18

Define Cox’s theorem

Answer 18

If Cox1,…,Cox 4 hold
then there is a continuous, strictly increasing surjective function g: [0,1] → [0,1]
such that g(

Question 19

Define the Betting Justification membership function XA of set A ⊆ W

Answer 19

XA : W -> {0,1}

Question 20

Define Bet1 of the Betting Justification

Answer 20

Gain S(1-p) if A is true and lose Sp if A is false

Question 21

Define Bet2 of the Betting Justification

Answer 21

Lose S(1-p) if A is true and gain Sp if A is false

Question 22

Under the Betting Justification what is the agents gain if they pick Bet 1?

Answer 22

S(1-p)XA(w) - Sp(1-XA(w)) = S(XA(w*)-p)

Question 23

Under the Betting Justification what is the agents gain if they pick Bet 2?

Answer 23

Sp(1-XA(w)) - S(1-p)XA(w) = -S(XA(w*)-p)

Question 24

How many values must be defined to specify a joint probability distribution for n binary variables?

Answer 24

2n - 1

Question 25

Defines the probability distribution for P(X1 = x1|X2 = x2) assuming X1 and X2 are dependent

Answer 25

P(X1=x1, X2=x2) / P(X2=x2)

Question 26

Define the probability distribution for P(X1 = x1 | X2= x2) assuming X1 is independent of X2

Answer 26

P(X1=x1)

Question 27

How many values must be defined to specify a joint probability distribution for n binary variables?

Answer 27

n

Question 28

What is the implicit assumption when Conditional Independence is used?

Answer 28

Some information is irrelevant