Binomial
Number of successes in n independent trials with probability p. Mean np, variance np(1-p).
Arises whenever you count successes in a fixed number of attempts.
Poisson
Number of events in a fixed interval when they occur independently at constant rate lambda. Mean and variance are both lambda, which is its distinguishing feature and a common interview check.
Arises as the limit of a binomial with large n, small p, and np held at lambda - so it is the distribution of rare events among many opportunities. Order arrivals in a market are commonly modelled this way.
Exponential
Waiting time until the next Poisson event. Mean 1/lambda, and memoryless - see waiting times.
The discrete analogue is the geometric distribution.
Normal
Arises from summing many independent contributions, via the central limit theorem. Symmetric, thin-tailed, fully described by mean and variance.
Worth memorising: about 68% within one standard deviation, 95% within two, 99.7% within three.
Lognormal
X is lognormal if log X is normal. Positive, right-skewed.
Prices are modelled this way because returns compound multiplicatively - the sum of many small log-returns is normal by the CLT, so the price itself is lognormal. It also enforces positivity, which prices require.
Note E[X] = exp(mu + sigma^2 / 2), not exp(mu). That gap between the mean and the median catches people, and it is why the expected value of a lognormal exceeds its median.
The relationships to know
- Binomial goes to Poisson when events are rare and trials many.
- Binomial goes to normal when trials are many and p is moderate.
- Sum of exponentials is gamma; sum of squared normals is chi-squared.
- Ratio of a normal to a chi-squared root gives Student's t - heavier tails, which is why it is used for returns.
The interview form
Questions are usually "which distribution and why", not "state the density". Explaining that arrivals are Poisson because events are independent at a constant rate is the answer being looked for.
Practise in random variables.