Theory Of Point Estimation Solution Manual -

Taking the logarithm and differentiating with respect to $\mu$ and $\sigma^2$, we get:

$$\frac{\partial \log L}{\partial \mu} = \sum_{i=1}^{n} \frac{x_i-\mu}{\sigma^2} = 0$$ theory of point estimation solution manual

$$\frac{\partial \log L}{\partial \lambda} = \sum_{i=1}^{n} \frac{x_i}{\lambda} - n = 0$$ Taking the logarithm and differentiating with respect to

$$L(\mu, \sigma^2) = \prod_{i=1}^{n} \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x_i-\mu)^2}{2\sigma^2}\right)$$ theory of point estimation solution manual

$$\hat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i-\bar{x})^2$$