nbinstat

Negative binomial mean and variance

Syntax

[M,V] = nbinstat(R,P)

Description

[M,V] = nbinstat(R,P) returns the mean of and variance for the negative binomial distribution with corresponding number of successes, R and probability of success in a single trial, P. R and P can be vectors, matrices, or multidimensional arrays that all have the same size, which is also the size of M and V. A scalar input for R or P is expanded to a constant array with the same dimensions as the other input.

The mean of the negative binomial distribution with parameters r and p is rq / p, where q = 1 – p. The variance is rq / p2.

The simplest motivation for the negative binomial is the case of successive random trials, each having a constant probability P of success. The number of extra trials you must perform in order to observe a given number R of successes has a negative binomial distribution. However, consistent with a more general interpretation of the negative binomial, nbinstat allows R to be any positive value, including nonintegers.

Examples

p = 0.1:0.2:0.9;
r = 1:5;
[R,P] = meshgrid(r,p);
[M,V] = nbinstat(R,P)
M =
9.0000  18.0000  27.0000  36.0000  45.0000
2.3333  4.6667  7.0000  9.3333  11.6667
1.0000  2.0000  3.0000  4.0000  5.0000
0.4286  0.8571  1.2857  1.7143  2.1429
0.1111  0.2222  0.3333  0.4444  0.5556

V =
90.0000 180.0000 270.0000 360.0000 450.0000
7.7778  15.5556  23.3333  31.1111  38.8889
2.0000  4.0000  6.0000  8.0000  10.0000
0.6122  1.2245  1.8367  2.4490  3.0612
0.1235  0.2469  0.3704  0.4938  0.6173