Negative Binomial distribution

Introduction to Negative Binomial Distribution

  • A Negative Binomial distribution is a discrete probability distribution found in statistical mathematics.
  • This distribution is the probability of a specified number of successful outcomes in a sequence of independent and identically distributed Bernoulli trials before a predetermined number of failures occurs.

Characteristics of Negative Binomial Distribution

  • This type of distribution only takes integer values, covering the range from 0 to infinity.
  • Just like a Bernoulli trial, each trial in a Negative Binomial Distribution is independent of others.
  • Each trial has only two possible outcomes: “success” or “failure”.
  • The probability of success (p) remains constant from trial to trial.

Using Negative Binomial Distribution

  • The formula for Negative Binomial distribution is P(X=x) = C(x-1, r-1) * p^r * (1-p)^(x-r) where x is the number of trials, r is the number of successes, and p is the probability of success.
  • This distribution can be used to model the number of trials needed to get a certain amount of successes or to ascertain the probability of getting a certain number of successes in a given amount of trials.

Expectation and Variance of Negative Binomial Distribution

  • The expectation, or mean, of a Negative Binomial distribution is r/p.
  • The variance of this distribution is r*(1-p)/p^2, where r is the number of successes and p is the probability of success.

When to Apply Negative Binomial Distribution

  • Use Negative Binomial distribution when dealing with a sequence of independent experiments, known as Bernoulli trials, where there are only two possible outcomes (a success or a failure), and the probability of success (p) remains constant.
  • It is often applied in quality control, reliability testing, and insurance claim modelling amongst others.

Further Points

  • Negative Binomial distribution is an advanced topic in the Further Stats 1 part of the maths Further Maths syllabus.
  • Grasping the concepts of Negative Binomial distribution is essential for understanding more elaborate statistical concepts such as overdispersion, which happens when observed variance in a data set exceeds what would be expected based on a certain statistical model.