Derive the negative binomial (NB) distribution.

(a) Suppose \(y\) follows a Poisson \((\lambda)\), where the parameter \(\lambda\) itself is a random variable following a gamma distribution \(\operatorname{Gamma}(p, r)\). Derive the distribution of \(y\). (Note that the density function of a \(\operatorname{Gamma}(p, r)\) is \(\frac{\lambda^{r-1} \exp (-\lambda p /(1-p))}{\Gamma(r)((1-p) / p)^{r}}\) for \(\lambda>0\) and 0 otherwise.)

(b) Derive the distribution of the number of trials needed to achieve \(r\) successes, where each trial is independent and has the probability of success \(p\). Compare it with the distribution in part (a).