The Distribution Zoo: Which One and Why

Interviewers rarely say "use the Poisson distribution." They describe a situation and watch whether you reach for the right model. This lesson is the matching table, with each distribution's two or three load-bearing facts.

Counting successes: Bernoulli → Binomial

$n$ independent yes/no trials, each succeeding with probability $p$: successes are Binomial($n, p$) with

$$E = np, \qquad \operatorname{Var} = np(1-p).$$

The variance formula is the quietly important one: 100 fair coin flips have mean 50, SD $\sqrt{25} = 5$ — the numbers behind every "make me a market on the number of heads" question. For large $n$, the normal approximation kicks in (CLT), and the tail ladder does the rest.

Rare events: Poisson

Events arriving independently at average rate $\lambda$ per interval: the count in one interval is Poisson($\lambda$):

$$P(k) = \frac{\lambda^k e^{-\lambda}}{k!}, \qquad E = \operatorname{Var} = \lambda.$$

Mean equals variance — the signature fact, and a diagnostic (count data with variance ≫ mean isn't Poisson; it's clustered). Poisson is the $n\to\infty$, $p\to 0$ limit of the binomial with $np = \lambda$ — so it models typos per page, trades per millisecond in a quiet name, market crashes per decade. Interview staple: "Orders arrive at 2 per second. Probability of a silent second?" — $e^{-2} \approx 13.5\%$.

Waiting: Geometric and Exponential

Geometric (discrete): trials until first success, $E = 1/p$. Exponential (continuous): waiting time at rate $\lambda$, $E = 1/\lambda$, density $\lambda e^{-\lambda t}$. Both share the property that generates the trick questions — memorylessness:

$$P(T > s + t \mid T > s) = P(T > t)$$

Having waited ten minutes for a bus with exponential arrivals tells you nothing; expected remaining wait is unchanged. Any question of the form "you've already flipped 5 tails, how many more flips until heads?" is testing whether you know (and can justify) this. Exponential gaps ↔ Poisson counts: two views of the same arrival process.

Sizes and prices: Normal and Lognormal

Normal: sums of many small independent effects (CLT). Working numbers: 68/95/99.7 within 1/2/3σ.

Lognormal: $e^{\text{normal}}$ — what you get when returns are normal and compound. Prices are modeled lognormal because they multiply and can't go negative. Two facts with teeth: lognormal is right-skewed, and its mean exceeds its median ($e^{\mu + \sigma^2/2}$ vs $e^{\mu}$) — the same $\sigma^2/2$ volatility-drag term from the calculus lesson. "Why can't stock prices be normal?" — negative prices and multiplicative dynamics; one sentence each.

Picking fast

Situation	Distribution
Count of successes in fixed trials	Binomial
Count of rare events in an interval	Poisson
Trials / time until first event	Geometric / Exponential
Sum or average of many small effects	Normal
Compounded growth, prices	Lognormal
"Completely unknown in a range"	Uniform

The interview version

"A desk gets on average 3 client calls an hour. What's the chance of exactly 5 next hour, and what assumption are you making?" — Poisson: $\frac{3^5 e^{-3}}{5!} \approx 10\%$; assuming independent, non-clustered arrivals — which client calls famously aren't (news clusters them), and saying so is the second half of the answer. Every distribution question has this two-part structure: compute under the model, then name the model's weak spot.