FRM distributions

SECTION 1: THE DISCRETE FOUNDATION

• Bernoulli: Single trial. E[X] = p, Var = p(1-p). Binary (Default/No Default).

• Binomial: n trials. E[X] = np, Var = np(1-p). Counts defaults in a pool. Assumes independence (Major exam trap: Contagion violates this).

• Poisson: Rare events in time. E[X] = \lambda, Var = \lambda. Used for operational risk/cyber-attacks.

• Poisson Scaling: If \lambda is annual, divide by 12 for monthly. \lambda_{total} = \sum \lambda_i.

• Poisson Approximation: n > 100 and np < 10.

SECTION 2: THE CONTINUOUS CORE

• Normal Distribution: Symmetrical. Skew=0, Kurtosis=3, Excess Kurtosis=0. Models asset returns.

• Lognormal Distribution: Bounded at 0, Positively skewed. Models asset prices.

• Lognormal Parameters: Mean = e^{\mu + 0.5\sigma^2}; Variance = [e^{\sigma^2} - 1] \cdot e^{2\mu + \sigma^2}. (High difficulty trap).

• Student’s t: Bell-shaped, heavier tails (Leptokurtic). Used for small samples (n < 30) or unknown variance.

• t-Dist Property: Var = df / (df - 2) for df > 2. As df \to \infty, it becomes Normal.

SECTION 3: COMPARISON & TRANSFORMATIONS

• Chi-Square (\chi^2): Sum of n squared standard normals. E[X] = df, Var = 2df. Tests variance of one sample.

• F-Distribution: Ratio of two Chi-Squares. Tests if two variances are equal. Always put larger variance in numerator.

• Central Limit Theorem (CLT): Distribution of sample means \to Normal as n \uparrow even if the underlying data is skewed.

• Transformation: Normal \to Lognormal (e^X). Normal \to Chi-Square (Z^2).

SECTION 4: THE "BOSS" LEVEL STRATEGY

• Decision Tree (Discrete): Count of fixed trials = Binomial. Count of rate over time = Poisson.

• Decision Tree (Continuous): Prices/Variables that can't be negative = Lognormal. Returns = Normal. Fat Tails = Student's t.

• Kurtosis Traps: Leptokurtic (Kurt > 3) = Underestimates VaR. Platykurtic (Kurt < 3) = Overestimates VaR.

• Z-Table Essentials: 95% (1-tail) = 1.645. 99% (1-tail) = 2.33. 95% (2-tail) = 1.96.

• Significance vs. Confidence: 5% Significance level = 95% Confidence level.