Consider a sequence of independent Bernoulli trials, each with same success probability $p$, where $0<p<1$. We continue performing trials until the *first* success.

Let $X$ be the number of failures before the first success. Then $X$ has the Geometric distribution with parameter $p$. The textbook writes $X∼Geom(p)$.

## PMF

The PMF of $X$ for a support $k$ is given by

$P(X=k)=q_{k}p$Where $k∈[0,1,2,...]$ and $q=1−p$. Intuitively, this says that we fail $k$ times, each with probability $1−p$ (since $p$ is success rate), and then we finally succeed once.

## Expectation

Given that $X∼Geom(p)$, then $E(X)=p1−p $. 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. $X$. The textbook doesn’t, but instead defines a new distribution.

If $X∼Geom(p)$, then the **First Success distribution** is given by $Y=X+1$, and the textbook writes this as $Y∼FS(p)$ with the same success rate $p$ as $X$.

### Expectation

Furthermore, the First Success expectation is then given by

$E(Y)=E(X+1)=E(X)+1=p1−p +1=p1 $