μ

G_{N} (s) = e^{μ (s - 1)}

. The preceding part gives

G_{F} (s) = e^{μ p (s - 1)}

, the probability generating function of a Poisson distribution with parameter

μ p

. To prove independence, retain both counts:

E [s^{F} t^{S} ∣ N] = (p s + (1 - p) t)^{N} .

(1)

Consequently the joint probability generating function factors as

E [s^{F} t^{S}] = e^{μ (p s + (1 - p) t - 1)} = e^{μ p (s - 1)} e^{μ (1 - p) (t - 1)} .

(2)

Comparing the coefficients of

s^{f} t^{k}

proves that the joint probabilities are the products of the marginal probabilities. Hence $F$ and $S$ are independent, with distributions $Poisson (μ p)$ and $Poisson (μ (1 - p))$ . This is Poisson thinning, including the degenerate counts when