Negative binomial distribution

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A negative binomial experiment is a statistical experiment that has the following properties:

  • The experiment consists of x repeated trials
  • Each trial can result in just two possible outcomes: success or failure
  • The probability of success, denoted by P, is the same on every trial
  • The trials are independent: that is, the outcome on one trial does not affect the outcome on other trials
  • The experiment continues until r successes are observed, where r is specified in advance.

Consider the following statistical experiment. You flip a coin repeatedly and count the number of times the coin lands on heads. You continue flipping the coin until it has landed 5 times on heads. This is a negative binomial experiment because:

  • The experiment consists of repeated trials. We flip a coin repeatedly until it has landed 5 times on heads.
  • Each trial can result in just two possible outcomes - heads or tails
  • The probability of success is constant - 0.5 on every trial
  • The trials are independent; that is, getting heads on one trial does not affect whether we get heads on other trials.
  • The experiment continues until a fixed number of successes have occurred; in this case, 5 heads.

Notation