The likelihood function is given by:
$$L(\lambda) = \prod_{i=1}^{n} \frac{\lambda^{x_i} e^{-\lambda}}{x_i!}$$ theory of point estimation solution manual
Solving this equation, we get:
$$\hat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i-\bar{x})^2$$ The likelihood function is given by: $$L(\lambda) =