Bernoulli trials until first success

Definition

In a sequence of independent Bernoulli trials with \(P(success) = \pi\) in each trial, the number of trials until the first success is observed has a distribution called a geometric distribution.

\[ X \;\; \sim \; \; \GeomDistn(\pi) \]

The probability function of a geometric random variable is relatively simple.

Probability function

If a random variable has a geometric distribution, \(X \sim \GeomDistn(\pi) \), then its probability function is

\[ p(x) = \pi (1-\pi)^{x-1} \quad \quad \text{for } x = 1, 2, \dots \]

(Proved in full version)

The geometric probability function can be directly shown to satisfy the required properties of a valid probability function.

Negative probabilities are impossible
Since \(0 \le \pi \le 1 \), it must also be true that \(0 \le (1-\pi) \le 1 \). Therefore
\[ p(x) = \pi (1-\pi)^{x-1} \ge 0 \quad \quad \text{for all } x \]
The probabilities sum to one
\[ \sum_{x=1}^\infty {p(x)} \;=\; \sum_{x=1}^\infty {\pi (1-\pi)^{x-1}} \;=\; 1 \]

The second property can be proved using following mathematical result.

Sum of geometric series

If \(-1 < a < 1\), then

\[ \sum_{x=0}^\infty {a^x} = \frac 1 {1-a} \]

(Proved in full version)

Shape of the geometric distribution

Each geometric probability is \( (1-\pi) \) times that of the previous one, so the probabilities decrease steadily from the mode at \(x = 1\).