If we observe data coming from a distribution with known form but unknown parameters, estimating those parameters is our primary aim. If the distribution is uniform on $[0,\theta]$ with $\theta$ unknown, we already looked at two methods to estimate $\theta$ given $n$ i.i.d. observations $x_1,\dots,x_n$:

- Maximum likelihood, which maximizes the likelihood function and gives $\max\{ x_i\}$
- Moment estimator $2\sum x_i/n$, or twice the mean

The uniform distribution was an interesting example because maximum likelihood and moments gave two different estimates. But what about the Poisson distribution? It is supported on the natural numbers, depends on a single parameter $\mu$, and has density function

$$f(n) = \frac{e^{-\mu}\mu^n}{n!}$$

What about the two methods of parameter estimation here? Let’s start with the method of moments. It is easy to compute the moments of the Poisson distribution directly, but let’s write down the moment generating function of the Poisson distribution.

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