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;?) = c(?)e?k/?, k = 0,1,2,3

...where ? is the temperature of the source and c(?) 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(?)

.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 ?, based on K1

....,Kn, the numbers of photons emitted when the source is triggered n times

.4

. Show that the ML estimator is consistent

.