Question 1
Sample Space & Events
- Sample space: \(\Omega\)
- Event: \(A \subseteq \Omega\)
Probability & Random Variables
17 questions. Use Show Answer, then slide right (or use Next) to continue.
Card 1 of 17
Question 2
Probability Axioms
- Non-negativity: \(P(A) \ge 0\)
- Normalization: \(P(\Omega) = 1\)
- Additivity (disjoint \(A_i\)): \(P(\bigcup A_i) = \sum P(A_i)\)
Question 3
Union & Intersection
- Union (A ∪ B): A happens or B happens (or both).
- Intersection (A ∩ B): A and B happen together.
Question 4
Probability Rule for P(A ∪ B)
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Explanation: subtract the overlap to avoid double-counting.
Special case (disjoint): if \(A \cap B = \varnothing\), then \(P(A \cup B) = P(A) + P(B)\).
Question 5
Conditional Probability
\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \]
valid when \(P(B) > 0\).
Plain meaning: probability of A after B is known to have happened.
Question 6
Law of Total Probability
If {Bi} is a partition of Ω, then:
\[ P(A) = \sum P(A \mid B_i) P(B_i) \]
Plain meaning: weighted average over hidden/unobserved scenarios.
Question 7
Law of Total Probability (Example)
Prime: 0.6, \(P(\mathrm{Default} \mid \mathrm{Prime}) = 0.02\)
Subprime: 0.4, \(P(\mathrm{Default} \mid \mathrm{Subprime}) = 0.10\)
\[ P(\mathrm{Default}) = 0.6(0.02) + 0.4(0.10) = 0.052 \]
Question 8
Bayes’ Theorem
\[ P(B \mid A) = \frac{P(A \mid B) P(B)}{P(A)} \]
Plain meaning: update beliefs about B after observing A.
Question 9
Independence of Events
A and B are independent if:
\[ P(A \cap B) = P(A)P(B) \]
Equivalent: \(P(A \mid B) = P(A)\)
Question 10
Random Variable Definition
Random variable: \(X : \Omega \to \mathbb{R}\)
Plain meaning: maps outcomes to numbers.
Question 11
Discrete vs Continuous Random Variables
- Discrete: countable values (e.g., number of defaults).
- Continuous: values over intervals (e.g., returns, time, price).
Question 12
PMF, PDF, and CDF
- PMF (discrete): \(P(X = x)\)
- PDF (continuous): \(f(x)\)
- CDF: \(F(x) = P(X \le x)\)
Question 13
CDF ↔ PDF Relationship
- From PDF to CDF: \(F(x) = \int_{-\infty}^{x} f(t) \, dt\)
- From CDF to PDF (if differentiable): \(f(x) = \frac{d}{dx}F(x)\)
- Discrete: \(F(x) = \sum_{t \le x} P(X = t)\)
Question 14
Joint, Marginal, and Conditional Distributions
- Joint: \(f(x,y)\)
- Marginal (integrate/sum out): \(f_X(x) = \int f(x,y) \, dy\)
- Conditional: \(f(x \mid y) = \frac{f(x,y)}{f_Y(y)}\)
Question 15
Expectation & Variance
- \(\mathbb{E}[X] = \sum xP(x)\) or \(\mathbb{E}[X] = \int x f(x) \, dx\)
- \(\mathrm{Var}(X) = \mathbb{E}[(X - \mathbb{E}[X])^2]\)
Plain meaning: expectation = average, variance = spread.
Question 16
Covariance & Correlation
- \(\mathrm{Cov}(X,Y) = \mathbb{E}[(X-\mu_X)(Y-\mu_Y)]\)
- \(\mathrm{Corr}(X,Y) = \frac{\mathrm{Cov}(X,Y)}{\sigma_X \sigma_Y}\)
Plain meaning: covariance = co-movement; correlation = scaled strength (−1 to 1).
Question 17
LLN vs CLT
- LLN: sample average converges to the true mean.
- CLT: distribution of the sample mean converges to Normal (under standard conditions).
Key line: LLN = convergence; CLT = distribution.