Statistical Sampling

Geometric distribution

Consider an experiment which can be performed over and over again, and which has a probability p of success at any attempt, regardless of how many experiments has been performed before and what their outcomes were. The, if one defined the number of experiments N that must be performed to in the end have a successful attempt, N follows a geometric distribution.

Since all trials are independent of each other, and the probability of failure is 1 - p, the probability that N = n experiments must be performed to end on a success, is P(N = n) = (1 - p)^{n - 1} p (for n = 1, 2, 3, \dots).

Basic properties:
Notation:
N \sim \mathrm{G}(p)
Type:
Discrete
Parameters:
p \in [0, 1]
Variables:
N
Support:
\mathbb{N} = \{ 1, 2, 3, \dots \}
PDF:
f(n) = P(N = n) = (1 - p)^{n - 1} p
CDF:
F(n) = P(N \le n) = 1 - (1 - p)^n
Mean:
\frac{1}{p}
Variance:
\frac{1 - p}{p^2}
Generating function:
\frac{tp}{1 - t (1 - p)}