http://www.ck12.org Chapter 4. Discrete Probability Distribution
4.6 The Geometric Probability Distribution
Learning Objectives
- Know the definition of the Geometric distribution.
- Identify the characteristics of the Geometric distribution.
- Identify the type of statistical situation to which the Geometric distribution can be applied.
- Use the Geometric distribution to solve statistical problems.
Like the Poisson and binomial distributions, the geometric distribution describes a discrete random variable. Recall,
in the binomial experiments, that we tossed the coin a fixed number of times and counted the number,x, of heads as
successes.
The geometric distribution describes a situation in which we toss the coin until the first head (success) appears. We
assume, as in the binomial experiments, that the tosses are independent of each other. The box below lists the main
characteristics of the geometric random variable.
Characteristics of the Geometric Probability Distribution
- The experiment consists of a sequence of independent trials.
- Each trial results in one of two outcomes: Success(S)or Failure(F).
- The geometric random variablexis defined as the number of trials until the firstSis observed.
- The probabilityp(x)is the same for each trial.
Why do we wait until a success is observed? For example, in the world of business, the business owner wants
to know the length of time a customer will wait for some type of service. Or, an employer, who is interviewing
potential candidates for a vacant position, wants to know how many interviews he/she has to conduct until the
perfect candidate for the job is found. Or, a police detective might want to know the probability of getting a lead in
a crime case after 10 people are questioned.
Probability Distribution, Mean, and Variance of a Geometric Random Variable
p(x) = ( 1 −p)x−^1 p x= 1 , 2 , 3 ,...μ=1
p
σ^2 =
1 −p
p^2where,