1. ## Method of Moments

The PMF of the Poisson distribution is given. E(X) = Lamda and Var(X) = lamda.

Find the method of moments estimator for: lamda with a "^" on top = lamda.

Sorry but I don't know how to add mathematical operators in the post so it would be easier for you to read!

2. In the poisson case you set the sample mean equal to the population mean getting...

$\hat\lambda =\bar X$

3. In general, with the method of moments, find the relationship between the moment ( $m_k=\mathbb{E}(X^k)$) and the parameter $\lambda$.

Let's say you have $\lambda=g(m_k)$

Then define $\hat m_k=\frac 1n \sum_{i=1}^n X^k$

You'll have $\hat\lambda=g(\hat m_k)$ and that's all...

Sometimes, you may have a multivariable function : $\lambda=g(m_k,m_p,\dots)$

In which case, it's exactly the same reasoning.