If X∼Negative Binomial(r,p)X \sim \text{Negative Binomial}(r, p)X∼Negative Binomial(r,p) (number of failures before rrr successes), what is the MGF MX(t)M_X(t)MX(t)?
(p1−(1−p)et)r(\frac{p}{1-(1-p)e^t})^r(1−(1−p)etp)r
(1−p1−pet)r(\frac{1-p}{1-pe^t})^r(1−pet1−p)r
p1−(1−p)et\frac{p}{1-(1-p)e^t}1−(1−p)etp
ert(1−p)/pe^{rt(1-p)/p}ert(1−p)/p