A r.v. has the Bernoulli distribution with a probability if and .

  • Note that cannot be exactly or , so both of these probabilities are positive.
  • We say that has the Bernoulli distribution. The textbook writes it as .

Bernoulli trial

An experiment represented by an Indicator variable for some event that we’re testing.

  • if occurs, and equals 0 if doesn’t occur.
  • If , then , where .


Since can only take on a value of or , the PMF of can be given, for , by


If is an indicator variable with , then its Variance .

We can prove this by showing , because the support and so . We have .

The largest possible variance of an indicator r.v. is , if .


Use LOTUS to calculate the MGF. We have

for any .