a. Assume that \(y_{1}, \ldots, y_{n}\) are i.i.d. with a negative binomial distribution with parameters \(r\) and \(p\). Determine the maximum likelihood estimators.

b. Use the sampling mechanism in part (a) but with parameters \(\sigma=1 / r\) and \(\mu\) where \(\mu=r(1-p) / p\). Determine the maximum likelihood estimators of \(\sigma\) and \(\mu\).

c. Assume that \(y_{1}, \ldots, y_{n}\) are independent with \(y_{i}\) having a negative binomial distribution with parameters \(r\) and \(p_{i}\), where \(\sigma=1 / r\) and \(p_{i}\) satisfies \(r\left(1-p_{i}\right) / p_{i}=\exp \left(\mathbf{x}_{i}^{\prime} \boldsymbol{\beta}\right)\left(=\mu_{i}\right)\). Determine the score function in terms of \(\sigma\) and \(\boldsymbol{\beta}\).