CONTENTS

Solution: Sum of Two Independent Poissons via CFs

Exercise: Sum of Two Independent Poissons via CFs

Part 1

φNi(t)=exp⁡(λi(eit−1))\varphi_{N_i}(t) = \exp(\lambda_i(e^{it} - 1)). By CF multiplicativity (independence):

φN(t)=φN1(t)φN2(t)=exp⁡((λ1+λ2)(eit−1)).\varphi_N(t) = \varphi_{N_1}(t)\varphi_{N_2}(t) = \exp((\lambda_1 + \lambda_2)(e^{it} - 1)).

This is the CF of a Poisson(λ1+λ2)\text{Poisson}(\lambda_1 + \lambda_2). By uniqueness, N∼Poisson(λ1+λ2)N \sim \text{Poisson}(\lambda_1 + \lambda_2).

Part 2

With N1=N2=N∗∼Poisson(λ)N_1 = N_2 = N^* \sim \text{Poisson}(\lambda), N1+N2=2N∗N_1 + N_2 = 2N^*. Its CF:

φ2N∗(t)=E[ei⋅2t⋅N∗]=φN∗(2t)=exp⁡(λ(e2it−1)).\varphi_{2N^*}(t) = \mathbb{E}[e^{i\cdot 2t \cdot N^*}] = \varphi_{N^*}(2t) = \exp(\lambda(e^{2it} - 1)).
This is not the CF of a Poisson distribution — a Poisson with rate μ\mu has CF exp⁡(μ(eit−1))\exp(\mu(e^{it} - 1)), not exp⁡(μ(e2it−1))\exp(\mu(e^{2it} - 1)). What is 2N∗2N^*? A scaled Poisson that takes only even values. It has the same support structure as Poisson but double the spacing — a "lattice-2" distribution. The lesson: the Poisson convolution rule requires independence.

Part 3 — Extension

Given N1+N2=nN_1 + N_2 = n, condition and compute:

P(N1=k∣N1+N2=n)=P(N1=k)P(N2=n−k)P(N1+N2=n)=(nk)(λ1λ1+λ2)k(λ2λ1+λ2)n−k.\mathbb{P}(N_1 = k \mid N_1 + N_2 = n) = \frac{\mathbb{P}(N_1 = k)\mathbb{P}(N_2 = n - k)}{\mathbb{P}(N_1 + N_2 = n)} = \binom{n}{k}\left(\frac{\lambda_1}{\lambda_1 + \lambda_2}\right)^k\left(\frac{\lambda_2}{\lambda_1 + \lambda_2}\right)^{n-k}.

So N1∣N1+N2=n∼Binomial(n,λ1/(λ1+λ2))N_1 \mid N_1 + N_2 = n \sim \text{Binomial}(n, \lambda_1/(\lambda_1 + \lambda_2)). Given the total, the two Poissons allocate independently with the intensity-ratio probability — a fact used routinely in Poisson-process thinning.

Takeaways

  • Independence is essential for CF multiplicativity. φX+Y=φXφY\varphi_{X+Y} = \varphi_X\varphi_Y fails when XX and YY share structure.
  • Sums of independent Poissons are Poisson; products of (Poisson) CFs are Poisson CFs. This is the workhorse in any counting model — default arrivals, trade arrivals, order-book events.
  • Conditioned on the total, a bivariate Poisson becomes binomial. This is the infinitesimal content of the Poisson-process thinning property and underlies all sparse-signal modelling in microstructure.
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