Statistics

Let \(x_1, x_2, …x_n\) be i.i.d. random variables that follow the Poisson distribution with parameter \(\lambda\), i.e. \(f(x;\lambda)=\dfrac{e^{-\lambda}\lambda^x}{x!}\).

1. (10 points) Find the methods of moments estimator for \(E[x_i]=\lambda\).

2. (10 points) Is this estimate unbiased? (show your work)

3. (10 points) Write an R function that would yield Methods of moments estimate for sample mean of a random sample of iid Poisson variables.

   1. Argument: A vector of $X$ values

   2. Output: MM estimate of $E[x_i]=\lambda$