Negative binomial inverse cumulative distribution function

`X = nbininv(Y,R,P)`

`X = nbininv(Y,R,P)`

returns
the inverse of the negative binomial cdf with corresponding number
of successes, `R`

and probability of success in a
single trial, `P`

. Since the binomial distribution
is discrete, `nbininv`

returns the least integer `X`

such
that the negative binomial cdf evaluated at `X`

equals
or exceeds `Y`

. `Y`

, `R`

,
and `P`

can be vectors, matrices, or multidimensional
arrays that all have the same size, which is also the size of `X`

.
A scalar input for `Y`

, `R`

, or `P`

is
expanded to a constant array with the same dimensions as the other
inputs.

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, `nbininv`

allows `R`

to
be any positive value, including nonintegers.

How many times would you need to flip a fair coin to have a 99% probability of having observed 10 heads?

flips = nbininv(0.99,10,0.5) + 10 flips = 33

Note that you have to flip at least 10 times to get 10 heads. That is why the second term on the right side of the equals sign is a 10.

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