Lectures on Elementary Probability

Chapter 4
The Bernoulli Process
4.1 Bernoulli random variables
A Bernoulli random variable X is a random variable with values 0 or 1. Consider
a Bernoulli random variable with P [X = 1] = p and P [X = 0] = 1 − p. Then
and
µ X = 1p + 0(1 − p) = p, (4.1)
σ 2 X = (1 − p) 2 p + (0 − p) 2 (1 − p) = p(1 − p). (4.2)
A Bernoulli random variable is also called an indicator function (or sometimes
a characteristic function). Let E be the event that X = 1. Then the
random variable X indicates this event. Conversely, given an event E, there is
a random variable X that has the value 1 on any outcome in E and the value 0
on any outcome in the complement of E. This constructs an indicator function
for the event.
4.2 Successes and failures
Consider an infinite sequence X 1 , X 2 , X 3 , . . . of independent Bernoulli random
variables, each with P [X i = 1] = p. This is called a Bernoulli process. It
consists of an infinite sequence of independent trials, each of which can result
in a success or a failure. The random variable X i is 1 if there is a success on
the ith trial, and it is 0 if there is a failure on the ith trial.
By independence, the probability of a particular sequence of successes and
failures is given by a product. Thus, for instance, the probability
P [X 1 = 1, X 2 = 0, X 3 = 1, X 4 = 1, X 5 = 0] = p(1 − p)pp(1 − p) = p 3 (1 − p) 2 .
(4.3)
More generally, the probability of a particular sequence of k successes and n − k
failures is p k (1 − p) n−k .
23