Given $\ell(\mu, \sigma)=-n \log (\sigma)-\frac{n}{2} \log (2 \pi)-\frac{1}{2 \sigma^{2}} \sum_{i=1}^{n}\left(x_{i}- \mu ight)^{2}$, solve the following problems : (a) $\frac{\partial \ell(\mu, \sigma)}{\partial \mu}=0$, solve for $\mu$ given $\sigma$ fixed. (b) $\frac{\partial \ell(\mu, \sigma)}{\partial \sigma) =0$, solve for $\sigma$ given $\mu$ fixed. Given $\ell(\lambda)=n \log (\lambda)-\lambda \sum_{i=1}^{n} X_{i}$, solve for $\lambda$ given $\frac{\partial \ell(\lambda)}{\partial \lambda)=0$ Given $\ell(\lambda)=-n \lambda+\log (\lambda) \sum_{i=1}^{n} x_{i}-\sum_{i=1}^{n} \log \left(x_{i} ! ight)$, solve for $\lambda$ given $\frac{\partial \ell(\lambda) }{\partial \lambda)=0 $ Here $x !$ stands for the factorial of $X$. SP.PB.094