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 .
PMF
Since can only take on a value of or , the PMF of can be given, for , by
variance
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 .
MGF
Use LOTUS to calculate the MGF. We have
for any .