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Bernoulli and Binomial Distributions

Bernoulli: one trial

A Bernoulli random variable takes the value 1 with probability pp and 0 with probability 1−p1-p. It is the simplest non-trivial distribution and the atom the rest are built from.

The Bernoulli moments
E[X]=pVar(X)=p(1−p)E[X] = p \qquad \text{Var}(X) = p(1-p)

A single trial. The variance peaks at one half, where the outcome is least predictable.

The variance result is worth internalising. It is maximised at p=0.5p = 0.5, where it equals 0.250.25, and it vanishes at p=0p = 0 or 11. Maximum uncertainty sits at even odds, which is exactly why a market at 50/50 is the hardest to trade around.

Bernoulli variables are the indicator variables from linearity of expectation: X=1X = 1 if some event happened, 0 otherwise.

Binomial: nn independent trials

Count the successes in nn independent Bernoulli trials, each with probability pp:

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