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Negative Binomial Distribution

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QXL Stat Tools Tab > Distribution Fit/Calc > Distribution Calculator > Negative Binomial

The Negative Binomial distribution is a discrete probability distribution. It can be used with two mutually exclusive outcomes of a Bernoulli trial, typically labeled "Success" and "Failure". The negative binomial has two parameters. p is the probability of success in a single Bernoulli trial. r is the number of successes before the experiment is terminated. The Negative Binomial returns the probability of observing x trials before the threshold number of successes (r).

NegBinomialpdf

Notes:

The Geometric Distribution is a special case where r = 1.

The Poison distribution is related to the Negative Binomial where r approaches infinity.

Example

You would like to know the probability of flipping heads 2 times in 5 attempts.

p = .5 (probability of heads)

Threshold Number of Successes = 2 (Two heads)

X Value = 5 (5 attempts)

The analysis from Quantum XL is below.

  • The bar when X=1 is zero. Since we need 2 successes (heads) it isn't possible to obtain two heads in a single flip. Therefore, the bar has a probability = 0.

  • The bar when X=2, represents the probability of flipping heads two times in two flips (two successes in a row). The probability of that occurring is .25.

  • P(X<=5) is the probability of seeing at least two heads in 5 flips. The probability is 81.25%.

  • P(X=5) is the probability of seeing the second head on the 5th flip (exactly the 5th flip, not before or after).

  • If you sum the bars for X=1, X=2, X=3, X=4, and X=5 then it will equal P(X<=5).

NegBinomialExample

See Also