Trials not starting from 1 - Negative binomial

Question

Trying to tabulate the pmf and cdf of a negative binomial distribution. I have a probability of 0.75, and an success of 2, where the number of trials starts at 2 (x = 2, 3, ...).

I am using r (the language) to calculate and I'm aware that the definitions of the negative binomial are different from my x, but is it a case of starting the vector from 2 to make the distributions match or is there something more complicated? Is the code below going to ensure the distributions passed into the dnbinom and pnbinom functions match up with x?

df <- data.frame(X = 2:20, 
           pmf = dnbinom(x = 2:20, size, prob),
           cdf = pnbinom(q = 2:20, size, prob, lower.tail = TRUE))

The PMF that is outputted from this is:

x   2.000 3.000 4.000 5.000 6.000    7
pmf 0.105 0.035 0.011 0.003 0.001    0

Answer 1

I solved this by changing the indexing of the vectors passed into the dnbinom and pnbinom functions to start at 0 rather than 2.

df <- data.frame(x = 2:12, 
           pmf = dnbinom(x = 0:10, size, prob),
           cdf = pnbinom(q = 0:10, size, prob, lower.tail = TRUE))

This leads to the expected output:

x   2.000 3.000 4.000 5.000 6.000 7.000 8.000    9
pmf 0.562 0.281 0.105 0.035 0.011 0.003 0.001    0
cdf 0.563 0.844 0.949 0.984 0.995 0.999 1.000    1