Question

A radioactive substance emits α-particle in such a way that the number of emitted particles during an hour, N, follows a Poisson distribution with parameter λ. The particle counter, however, is somewhat unreliable in the sense that each emitted particle is detected with probability p (0 ≤ p ≤ 1), whereas it remains undetected with probability q = 1 − p. All particles are detected independently of each other. Writing X for the number of detected particles during an arbitrarily chosen hour we have with

{X|N=n} ∼Bin(n,p) N ∼ Pois(λ).

(i) Use the results for conditional means and variances to calculate E[X] and var[X].

(ii) Use probability generating functions to find the marginal probability function of X.

(iii) Hence, by direct calculation, verify your results for E[X] and var[X].

Answer 1