Properties
- A geometric distribution represents an experiment that consists of a random number of identical and independent trials denoted as x. The random variable is how many trials are needed before the first success.
- Each of the trials in the experiment has only two results - either a success or a failure, and each has a probability of success denoted as p.
What do these properties mean?
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Random Number of Trials
This means that the number of trials that are going to be conducted in the experiment is unknown before the experiment is over. Because this is the random variable, it means that it is the thing in the experiment that is likely to change from experiment to experiment.
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Identical and Independent
The trials being identical and independent means that the result of one trial has no effect on any other trials. This means that the probability of a success does not change trial to trial.
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Success or Failure
The definition of a success in a geometric experiment is that you got whatever you were looking for. Therefore, the definition of a failure in a geometric experiment is that you did not get whatever you were looking for.
Example
When dealing with an experiment, if you see or hear the words "before" or "until" you are probably dealing with a either a geometric or a negative binomial experiment. If the experiment is continued until a single thing is achieved, then it is a geometric experiment. If the experiment is continued until an amount of things other than one are achieved, it is a negative binomial experiment.
For example, imagine you have a standard six-sided die and you are interested in how many times you have to roll it before it turns up a six.
This would be a geometric distribution with p = 1/6 or about 0.1667 (because there are six sides on a die and only one is a success).
The only thing left would be to determine how many times you would like to roll the die, and that would be x (because it is the number of trials you would like to conduct).
Note that you will not always conduct exactly that number of trials - there is only some probability that you will.
Requirements