Key Concepts

• The probability of success and failure is the same in each trial of a geometric experiment.
• In a geometric experiment, $X =$ 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 $(RV)$ that arises from the Bernoulli trials; the trials are repeated until the first success. The geometric variable $X$ is defined as the number of trials until the first success. Notation: $X \sim G(p)$. The mean is $\mu = \frac{1}{p}$ and the standard deviation is $\sigma = \sqrt{\frac{1}{p} (\frac{1}{p} - 1)}$. The probability of exactly $x$ failures before the first success is given by the formula: $P(X=x)=p(1-p)^{x-1}$.

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, $p$, of a success and the probability, $q$, of a failure do not change from trial to trial.