What is a negative binomial distribution

negative binomial distribution

Pascal distribution, that by the discrete probability density

\ begin {eqnarray} f: {{\ mathbb {N}}} _ {0} \ ni k \ to \ left (\ begin {array} {c} m + k-1 \ m-1 \ end {array } \ right) {p} ^ {m} {(1-p)} ^ {k} \ in [0,1] \ end {eqnarray}

on the power set \ ({\ mathfrak {P}} ({{\ mathbb {N}}} _ {0}) \) of the natural numbers including zero and defined by the parameters m ∈ & Nopf; and p ∈ (0, 1) dependent discrete probability measure P.

The value P({k}) = f (k) the negative binomial distribution with the parameters m and p indicates the probability that, in the case of independent repetitions of a random experiment with the two outcomes, success and failure and the probability of success p before the m-th success exactly k Failures occur. The negative binomial distribution thus represents a generalization of the geometric distribution. The distribution owes its name to the fact that the probabilities of the elementary events in the form

\ begin {eqnarray} P (\ {k \}) = \ left (\ begin {array} {c} -m \ k \ end {array} \ right) {(- 1)} ^ {k} {( 1-p)} ^ {k} {p} ^ {m} \ end {eqnarray}

which is reminiscent of the binomial distribution.

Is X one with the parameters m and p negative binomial random variable, then applies to the expected value

\ begin {eqnarray} E (X) = m \ frac {1-p} {p} \ end {eqnarray}