4 A source emits a random number of photons K each time that it is triggered. We assume that the PMF of K is
pK(k;θ) = c(θ)e-k/θ, k = 0,1,2,3...
where θ is the temperature of the source and c(θ) is a normalization factor. We also assume that the photon emissions each time that the source is triggered are independent. We want to estimate the temperature of the source by triggering it repeatedly and counting the number of emitted photons.
1. Determine the normalization factor c(θ).
2. Find the expected value and the variance of the number K of photons emitted if the source is triggered once.
3. Derive the ML estimator for the temperature θ, based on K1....,Kn, the numbers of photons emitted when the source is triggered n times.
4. Show that the ML estimator is consistent.