Bernoulli distribution describes an outcome of a single experiment with two possible outcomes (success or failure): \(X=1\) if an experiment was successfull, \(X=0\) if it was not.

Parameter: \(p\) – probability of a success.

Values: \(\{0,1\}.\)

Probability mass function: \[ P(X=1)=p, \ P(X=0)=1-p. \]

Moment generating function: \[ M(t)=1-p+pe^t \]

Proof

\[ M(t)=Ee^{tX}=e^t\cdot P(X=1)+e^0\cdot P(X=0)=pe^t+1-p \]

All moments: \[ EX^n=p, \ n\geq 1. \]

Proof

Observe that \(X^n=X,\) for \(n\geq 1.\) Further, \[ EX=1\cdot P(X=1)+0\cdot P(X=0)=p+0=p \]

Expectation: \(EX=p\)

Variance: \(V(X)=p(1-p)\)