Key Concepts

• The probability of success and failure is the same in each trial of a geometric experiment.
• In a geometric experiment, [latex]X =[/latex] the number of independent trials until the first success.
• The number of trials in a geometric experiment is not fixed.

Glossary

geometric distribution: a discrete random variable [latex](RV)[/latex] that arises from the Bernoulli trials; the trials are repeated until the first success. The geometric variable [latex]X[/latex] is defined as the number of trials until the first success. Notation: [latex]X \sim G(p)[/latex]. The mean is [latex]\mu = \frac{1}{p}[/latex] and the standard deviation is [latex]\sigma = \sqrt{\frac{1}{p} (\frac{1}{p} - 1)}[/latex]. The probability of exactly [latex]x[/latex] failures before the first success is given by the formula: [latex]P(X=x)=p(1-p)^{x-1}[/latex].

geometric experiment: a statistical experiment with the following properties:

1. There are one or more Bernoulli trials with all failures except the last one, which is a success.
2. In theory, the number of trials could go on forever. There must be at least one trial.
3. The probability, [latex]p[/latex], of a success and the probability, [latex]q[/latex], of a failure do not change from trial to trial.