5.3 Common Univariate Distributions

Common Univariate Distributions

QA–3 catalogs workhorse distributions. FRM questions rarely ask you to derive densities from scratch; they ask which law fits a risk object, what moments equal what parameters, and how tails compare.

Bernoulli and Binomial

A Bernoulli(p) RV is 1 with probability p and 0 with probability 1 − p. Mean p, variance p(1 − p). Use it for a single default/no-default, breach/no-breach, or up-move/down-move indicator.

A Binomial(n, p) RV is the sum of n i.i.d. Bernoulli(p) trials: X = number of successes in n independent trials. PMF: P(X = k) = C(n, k) pᵏ (1−p)^{n−k} for k = 0, …, n. Mean np, variance np(1−p).

Worked binomial example

Ten independent counterparties each have PD = 2% under a conditional-independence assumption given the scenario. Number of defaults X ~ Binomial(10, 0.02). E[X] = 0.20, Var(X) = 10·0.02·0.98 = 0.196. P(X = 0) = (0.98)¹⁰ ≈ 0.817. P(at least one default) ≈ 0.183. If defaults were positively correlated, the true P(X ≥ 1) and especially P(X large) would typically be higher than this independent binomial implies—exam awareness point for credit portfolios.

Poisson

Poisson(λ) counts events in a continuum (time, portfolio) when events are rare and “independent” in small intervals. P(X = k) = e^{−λ} λᵏ / k!. Mean = variance = λ. For large n and small p with np = λ, Binomial(n, p) ≈ Poisson(λ).

Operational risk frequency modules often use Poisson (or negative binomial for over-dispersion). If a desk averages λ = 3 material incidents per year, P(X = 0) = e^{−3} ≈ 0.050, so about a 5% chance of a clean year under the model.

Uniform

Discrete uniform on {1, …, m} gives equal weight 1/m. Continuous Uniform(a, b) has PDF f(x) = 1/(b−a) on [a, b], mean (a+b)/2, variance (b−a)²/12. Uniforms appear as “no information” priors on a bounded parameter, as random numbers U ~ Uniform(0,1) for simulation (inversion method: X = F⁻¹(U)), and as simplified recovery assumptions in crude models.

Normal

Normal(μ, σ²) has PDF shaped by the familiar bell curve, CDF Φ for the standard case N(0,1). Mean μ, variance σ², skewness 0, kurtosis 3 (excess 0). Affine transforms of normals are normal: aX + b ~ N(aμ + b, a²σ²). Sums of independent normals are normal. Central limit theorem justifies normal approximations for aggregated risks under regularity conditions.

Risk uses: parametric VaR (VaR_α ≈ μ + σ z_α for P&L or loss conventions carefully signed), many sampling distributions, and Brownian building blocks. Limitation: thin tails versus real returns; normal understates extreme move frequency.

Worked normal VaR sketch

Daily P&L ~ N(0, 2²) in $m (σ = $2m). For a 5% left-tail P&L VaR using z_{0.05} ≈ −1.645, the quantile is 0 + 2(−1.645) = −$3.29m, so a loss VaR of about $3.29m. Changing to a heavier-tailed law raises that number without changing σ.

Lognormal

If ln X ~ N(μ, σ²) with X > 0, then X is lognormal. E[X] = exp(μ + σ²/2), Var(X) = [exp(σ²) − 1] exp(2μ + σ²). Right-skewed; median exp(μ) < mean when σ > 0. Classic for stock prices in geometric Brownian motion, and sometimes for severity of operational losses. Trap: parameters μ, σ are for ln X, not for X; quoting “μ = 10%” without saying which scale confuses mean returns with drifts of the log.

Chi-Squared, Student’s t, and F

Chi-squared with ν degrees of freedom (χ²_ν) is the distribution of Σᵢ Zᵢ² for i.i.d. standard normals Zᵢ. Mean ν, variance 2ν, support [0, ∞). Sample variances of normals connect to χ²: (n−1)s²/σ² ~ χ²_{n−1} under classical assumptions.

Student’s t_ν is (roughly) a normal divided by an independent root-χ² scale. Mean 0 (for ν > 1), variance ν/(ν−2) for ν > 2, excess kurtosis 6/(ν−4) for ν > 4—heavier tails than normal, approaching normal as ν → ∞. Risk systems sometimes use t for returns to fatten tails with one extra parameter ν.

F_{d₁,d₂} is (ratio of) scaled chi-squareds; used to test equality of variances and in regression ANOVA. Mean d₂/(d₂−2) for d₂ > 2.

Exponential and Beta

Exponential(λ) waiting time with PDF λ e^{−λx} for x > 0, mean 1/λ, variance 1/λ². Memoryless: P(T > s + t | T > s) = P(T > t). Useful for simple time-to-default or time-to-event models with constant hazard; unrealistic when aging or regimes matter.

Beta(α, β) lives on (0, 1) with flexible shapes (U-shaped, hump-shaped, skewed). Mean α/(α+β). Natural for random recovery rates, uncertain PDs, or mixing weights. A Beta(2, 8) prior on LGD concentrates toward lower loss-given-default; Beta(0.5, 0.5) is U-shaped toward extremes.

Mixture Distributions

A mixture draws a latent state (or parameter) from one distribution, then draws X from a conditional law. Example: with probability 0.95, daily return ~ N(0, 0.01²); with probability 0.05, return ~ N(0, 0.05²). The unconditional distribution has fatter tails than either single normal with the average variance—volatility clustering and jump days in reduced form.

Credit and operational models mix Poisson frequencies with heavy severity; market-risk engines mix Gaussians across regimes. Mixtures explain why matching only mean and variance (method of moments with a single normal) understates capital: the mixture’s kurtosis and tail quantiles exceed the matched normal’s.

Worked mixture tail idea

Equal mixture of N(−1, 1) and N(1, 1) is bimodal with variance 1 + 1 = 2 from the law of total variance (mean of variances plus variance of means). A single N(0, 2) matches mean and variance but not shape: probabilities near ±1 differ. Risk reports that rely only on σ miss that distinction.

Selection Discipline

Choose distributions by support (can X be negative?), tail needs, dependence on parameters you can estimate, and tractability. Then stress the assumption: replace normal with t, binomial independence with a mixture over p, exponential with a Weibull, and re-estimate capital. Distribution choice is a model risk decision, not a cosmetic one.