Common Distributions

A working quant carries a dozen distributions in their head not as trivia but as a map of how randomness is generated. The power is in the relationships: the same Bernoulli trial, aggregated three different ways, becomes the Binomial, the Poisson, or the Normal. Knowing which limit you are in tells you which formula is safe and where it will betray you. This is a tour of the core family and the arrows between them.

The discrete backbone: Bernoulli → Binomial → Poisson

Bernoulli($p$). The atom of discrete probability: $X \in \{0,1\}$ with $\mathbb{P}(X=1)=p$. Mean $p$, variance $p(1-p)$ (maximised at $p=\tfrac12$). Every yes/no event, a trade fills, a bond defaults, a coin lands heads, is Bernoulli.

Binomial($n,p$). The count of successes in $n$ independent Bernoulli($p$) trials: $\mathbb{P}(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \qquad \mathbb{E}[X] = np, \quad \operatorname{Var}(X) = np(1-p).$ The mean and variance follow instantly from linearity applied to a sum of $n$ independent Bernoullis. Binomial models defaults in a homogeneous credit pool, up-moves in a binomial option tree, and any bounded count.

Poisson($\lambda$). The limit of Binomial as $n \to \infty$, $p \to 0$, with $np \to \lambda$ fixed, the law of rare events: $\mathbb{P}(X=k) = e^{-\lambda}\frac{\lambda^k}{k!}, \qquad \mathbb{E}[X] = \operatorname{Var}(X) = \lambda.$ The derivation: $\binom{n}{k}p^k(1-p)^{n-k} \to \frac{\lambda^k}{k!}e^{-\lambda}$ using $(1 - \lambda/n)^n \to e^{-\lambda}$. Its signature is mean equals variance, a testable prediction: if your count data has variance far above the mean it is overdispersed and Poisson is wrong (use negative binomial). Poisson models order arrivals, trade counts, jumps, and rare-default settings. Its defining structural property is that the waiting time between events is exponential, the bridge to continuous time.

The continuous core: Normal, Exponential, Lognormal

Normal($\mu, \sigma^2$). The attractor of the The Central Limit Theorem, with density $f(x) = \frac{1}{\sqrt{2\pi}\,\sigma}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right).$ Mean $\mu$, variance $\sigma^2$, skewness $0$, kurtosis exactly $3$. It is closed under linear combinations, a sum of independent normals is normal, with means and variances adding, which is why Gaussian assumptions propagate so conveniently through linear portfolios. The standard normal $Z \sim \mathcal{N}(0,1)$ is the reference; any normal is $\mu + \sigma Z$. Its fatal flaw for finance is thin tails: $\mathbb{P}(|Z| > 4) \approx 6\times 10^{-5}$, so it wildly underprices crashes.

Exponential($\lambda$). The continuous waiting time for a Poisson event: $f(x) = \lambda e^{-\lambda x}\ (x \ge 0), \qquad \mathbb{E}[X] = 1/\lambda, \quad \operatorname{Var}(X) = 1/\lambda^2.$ Its defining property is memorylessness: $\mathbb{P}(X > s + t \mid X > s) = \mathbb{P}(X > t)$. A component that has survived to time $s$ is as good as new, the only continuous distribution with this property, which makes it the natural (and sometimes dangerously optimistic) model for time-to-default or time-between-trades.

Lognormal. If $\ln Y \sim \mathcal{N}(\mu,\sigma^2)$ then $Y = e^X$ is lognormal, the Black–Scholes model for prices. Support is $(0,\infty)$ (prices can't go negative), it is right-skewed, and crucially its mean is not $e^\mu$: $\mathbb{E}[Y] = e^{\mu + \sigma^2/2}, \qquad \operatorname{Var}(Y) = \big(e^{\sigma^2}-1\big)e^{2\mu + \sigma^2}.$ The $+\sigma^2/2$ is a Jensen Expectation, Variance & Moments convexity term, the gap between the mean of the log and the log of the mean, and forgetting it is one of the most common pricing errors. Modelling returns as normal is equivalent to modelling prices as lognormal.

The web of relationships

These distributions are not independent facts; they are one structure viewed from different limits:

• Bernoulli sums → Binomial. $n$ i.i.d. Bernoulli($p$) add to Binomial($n,p$).
• Binomial → Poisson. Many trials, tiny success probability, moderate expected count.
• Binomial → Normal. Many trials, fixed $p$: de Moivre–Laplace, the original CLT. Rule of thumb: valid when $np$ and $n(1-p)$ both exceed ~10.
• Poisson → Normal. Large $\lambda$: a Poisson($\lambda$) is approximately $\mathcal{N}(\lambda,\lambda)$.
• Poisson ↔ Exponential. Event counts are Poisson; inter-event gaps are exponential; both are faces of the Poisson process.
• Exponential → Gamma. A sum of $k$ i.i.d. exponentials is Gamma($k, \lambda$) (Erlang), the waiting time for the $k$-th event.
• Normal → Chi-squared, $t$, $F$. Sums of squared standard normals give $\chi^2$; a normal over an independent $\sqrt{\chi^2/\nu}$ gives Student-$t$, whose heavy tails make it the default fat-tailed return model.

Worked example, Binomial-to-Poisson for defaults. A pool of $n = 1000$ loans each defaults independently with $p = 0.002$. The exact default count is Binomial($1000, 0.002$); computing $\mathbb{P}(X = k)$ with factorials of 1000 is ugly. The Poisson approximation with $\lambda = np = 2$ gives $\mathbb{P}(X = 0) = e^{-2} \approx 0.135$, $\mathbb{P}(X = k) = e^{-2}2^k/k!$, accurate to three decimals and computable by hand. This is exactly the regime (large $n$, tiny $p$) the Poisson limit was built for.

Failure modes and subtleties

• Normal tails are lethally thin. Real returns have excess kurtosis; a Gaussian VaR systematically under-reserves. Fat-tailed alternatives (Student-$t$, stable, jump-diffusion) exist for a reason.
• Poisson overdispersion. If event-count variance exceeds the mean, Poisson is misspecified; the negative binomial adds a mixing parameter.
• Memorylessness is often false. Default hazards rise with leverage and age; exponential's constant hazard understates clustering.
• Lognormal mean bias. The $e^{\sigma^2/2}$ factor means naive plug-in of the log-mean underprices; this compounds badly at high vol.
• Independence assumptions. Binomial and Poisson pool models assume independent events; correlated defaults (a common shock) fatten the tail far beyond either, the modelling failure at the heart of 2008 CDO losses.

In interviews

Know the mean and variance of each of these cold, they are the most common single-line questions. Be ready to derive the Binomial mean/variance from a sum of Bernoullis, and the Poisson from the Binomial limit. Expect "why does mean equal variance for Poisson, and what does it tell you if your data violates it?" A frequent one: "returns are normal, what does that say about prices?" (lognormal, with the $\sigma^2/2$ drift correction). Memorylessness of the exponential is a classic ("a bus arrives on average every 10 minutes; you've waited 5, how long until it comes?", still 10, in expectation). Their moment-generating functions, which make many of these relationships one-liners, are covered in Moment Generating Functions.