Bernoulli distribution
/ ber-NOO-lee /
The Bernoulli distribution is the simplest possible random experiment: a single yes-or-no event with just two outcomes. A coin lands heads or tails; a policyholder either makes a claim this year or does not; a patient survives the year or dies. Whenever the world hands you a single one-shot success-or-failure, you have a Bernoulli trial, and it is the atom from which many richer distributions are built.
A Bernoulli random variable X takes the value 1 (success) with probability p and 0 (failure) with probability 1 - p. That is all there is to it. Its mean is simply E[X] = p, and its variance is Var(X) = p(1 - p), which is largest when p = 0.5 (maximum uncertainty about a 50-50 event) and shrinks toward zero as p approaches 0 or 1 (near-certain outcomes carry little risk). The single parameter p is both the success probability and the average value.
In actuarial modeling the Bernoulli is the natural model for any one-period either/or risk. Does this single policy produce a claim? Does this one life die during the year? The one-year mortality rate q_x is exactly the success probability of a Bernoulli 'death' indicator. Because real portfolios contain many such independent yes/no trials, summing them turns Bernoulli into the binomial — which is why the Bernoulli is the foundation of frequency modeling and of the whole life table.
Model whether one 70-year-old dies this year as Bernoulli with p = q_70 = 0.02. The death indicator is 1 with probability 0.02, 0 otherwise; its mean is 0.02 and its variance is 0.02 × 0.98 = 0.0196.
A one-year death is a Bernoulli trial; the mortality rate q_x is its success probability.
A single Bernoulli trial tells you nothing reliable on its own — the value is always just 0 or 1; its usefulness appears only when many independent trials are pooled, which is the law of large numbers at work.