Consider a sequence of independent Bernoulli trials, each with same success probability , where . We continue performing trials until the first success.

Let be the number of failures before the first success. Then has the Geometric distribution with parameter . The textbook writes .


The PMF of for a support is given by

Where and . Intuitively, this says that we fail times, each with probability (since is success rate), and then we finally succeed once.


Given that , then . The proof is a bit gnarly (and there are like 5 of them) so we’ll prove it later in the semester.

First Success distribution

Some people define the Geometric distribution to include the first success within the r.v. . The textbook doesn’t, but instead defines a new distribution.

If , then the First Success distribution is given by , and the textbook writes this as with the same success rate as .


Furthermore, the First Success expectation is then given by