Suppose that X ∼ NB(r, p). In this formulation of the Negative Binomial distribution, X
represents the number of independent Bernoulli trials needed to obtain the r
thsuccess. To
obtain the limit theorem of interest here, it is useful to work with the pmf of the variable
Y = X −r, which represents the number of failures which occur before the r
th success in a
sequence of independent Bernoulli trials.
(a) Show that the pmf of the variable Y is given by
p(y) =
r +y−1
r −1
p
r
q
y
for y = 0,1,2,...
where q = 1− p.
(b) Show that P(Y = y) tends to λ
x
e
−λ
x!
, the pmf of the P(λ) distribution, as r → ∞, p → 1
and rq → λ > 0.