Introduction to Probability Theory for Risk Modeling

Question 1

State Space (Ω)

Answer 1

The set of all possible outcomes of an experiment.

Question 2

Event (A)

Answer 2

Any subset of the state space Ω.

Question 3

Probability Measure Axioms

Answer 3

P(A) ∈ [0,1], P(∅)=0, P(Ω)=1, and for disjoint A, B: P(A∪B)=P(A)+P(B).

Question 4

Complement Rule

Answer 4

P(A^c) = 1 - P(A).

Question 5

Union Rule

Answer 5

P(A∪B) = P(A) + P(B) - P(A∩B).

Question 6

Conditional Probability

Answer 6

P(A|B) = P(A∩B) / P(B), provided P(B)>0.

Question 7

Independence

Answer 7

A and B are independent if P(A∩B) = P(A)P(B).

Question 8

Bayes’ Theorem

Answer 8

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

Question 9

Random Variable (X)

Answer 9

A function X: Ω → ℝ assigning numerical values to outcomes.

Question 10

PMF (discrete)

Answer 10

p(x) = P(X = x).

Question 11

PDF (continuous)

Answer 11

f(x) such that P(a≤X≤b) = ∫ f(x) dx.

Question 12

CDF

Answer 12

F(x) = P(X ≤ x).

Question 13

Expectation (discrete)

Answer 13

E[X] = Σ x p(x).

Question 14

Expectation (continuous)

Answer 14

E[X] = ∫ x f(x) dx.

Question 15

Variance

Answer 15

Var(X) = E[X²] - (E[X])².

Question 16

Bernoulli Distribution

Answer 16

Models binary outcomes (success/failure)

Question 17

Binomial Distribution

Answer 17

Models multiple Bernoulli trials (e.g.,
default risk)

Question 18

Poisson Distribution

Answer 18

Models rare events over time (e.g., fraud
detection)

Question 19

Normal Distribution

Answer 19

Used for asset returns and risk modeling.

Question 20

Exponential Distribution

Answer 20

Models time between events (e.g.,
waiting time for defaults)

Question 21

Lognormal Distribution

Answer 21

Used for modeling stock prices