Properties
- A negative binomial distribution represents an experiment that consists of a random number of identical and independent trials denoted as x.
- 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.
- The trials continue until a fixed number of successes denoted as r are obtained.
- The random variable is how many trials are needed before the fixed number of successes are obtained.
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 negative binomial experiment is that you got whatever you were looking for. Therefore, the definition of a failure in a negative binomial 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 two ones. This would be a negative binomial distribution with p = 1/6 or about 0.1667 (because there are six sides on a die and only one is a success) and r = 2 (because you would like to roll two ones). 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
- The number of trials must be a whole number greater than zero and also at least equal to the number of successes (x >= r and x = 1, 2, 3 ...).
- The number of successes must be a whole number greater than or equal to zero (r = 0, 1, 2 ... x)
- The probability of success must be greater than or equal to zero and also less than or equal to one (1 >= p >= 0).