Common Distributions and Multivariate Random Variables

You model daily losses on a credit book as normal. The realized distribution has a -3 standard-deviation event every two months. The exam tests whether you know which distributions actually fit which risk problems, and how to handle two random variables together when one drives the other.

A Bernoulli trial has two outcomes with probability $p$. Mean = $p$, variance = $p(1-p)$. One bond either defaults or it doesn't.

A binomial random variable is the number of successes in $n$ independent Bernoulli trials each with the same probability $p$. The PMF is

$$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$

Mean = $np$, variance = $np(1-p)$. Use binomial when you have a fixed number of bonds with the same default probability and you want the distribution of total defaults. The independence assumption is restrictive: in stress, defaults cluster, so binomial understates tail risk.

KEY: Binomial assumes constant $p$ and independent trials. Real default behavior violates both: $p$ varies with the cycle and defaults are conditionally dependent.

Common mistakes

• Using normal for stock prices instead of returns. Prices cannot go negative, so price level is lognormal, not normal. Log-returns are normal under the standard assumption. Trap: a question models a stock price at $100 with $\sigma =\$30$ under "normality".
• Forgetting the convexity adjustment for the lognormal mean. $E[X]=e^{\mu +{\sigma }^2/2}$, not $e^{\mu }$. Trap: with $\mu =0.10$ and $\sigma =0.20$, the lognormal mean is $e^{0.12}=1.1275$, not $e^{0.10}=1.1052$. Skipping the half-variance is worth ~2% on a one-year horizon.
• Treating variance as additive across asset weights. Portfolio variance is $w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2+2w_1{w}_2\text{Cov}$. The trap answer drops the covariance term entirely. With two perfectly-correlated assets, that omission understates portfolio $\sigma$ by 30-50%.

Bottom line

• Bernoulli/Binomial for default counts, Poisson for arrival counts in fixed time, Normal for log-returns, Lognormal for price levels.
• Student t for fat-tailed returns, Chi-squared/F for variance and joint hypothesis tests, Exponential for inter-arrival times, Beta for [0,1] quantities.
• Mixture distributions combine component distributions weighted by mixing probabilities, the standard tool for modeling regime switches and leptokurtosis.
• Joint, marginal, conditional describe bivariate variables. Marginal sums (or integrates) out the other variable; conditional fixes one variable and renormalizes by the marginal.

Exam shortcut

When a question gives two variables and asks about their joint distribution, decide first whether independence applies. If yes, the joint factors into the product of marginals. If no, you need covariance, conditional structure, or a copula. The trap is assuming independence when correlations exist.

The full lesson (about 2,826 words, 19 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

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