In this question well use simulations to explore maximum likelihood estimation for the waiting time, , in

In this question we’ll use simulations to explore maximum likelihood estimation for the waiting time, λ, in a Poisson process.

Suppose we have k See Section 40.4.1 for more details. independent observations from the Exponential(λ) distribution. We can add these up and treat them as a single observation from the gamma distribution, X ∼

Gamma(k, λ).

In Section 40.3.5 we learnt that the maximum likelihood estimator of λ in this situation is λb= k X , and that its mean and variance are E(λb) = 
k k − 1 
λ, Var(λb) = k 2 λ
2 (k − 1)
2 (k − 2)
.
Write down R commands that you can use to simulate a sample of 107 values of λband find their empirical mean and variance, using the simulation settings below. Run the code in each case, and check your results for empirical mean and variance against the formulas given.

a. Simulation settings k = 5, λ = 1.2.

b. Simulation settings k = 100, λ = 1.2.
Based on these results, what do you notice about E(λb) and Var(λb) for the small sample size, k = 5, versus the large one, k = 100? Which sample size is better to use in practice if you want to estimate λ from your own data?