Solution: Sum of Independent Poissons via MGFs

Exercise: Sum of Independent Poissons via MGFs

Part 1

M_X(s) = \sum_{k=0}^\infty e^{sk}\,\frac{e^{-\lambda_1}\lambda_1^k}{k!} = e^{-\lambda_1}\sum_{k=0}^\infty \frac{(\lambda_1 e^s)^k}{k!} = e^{-\lambda_1}\,e^{\lambda_1 e^s} = \exp(\lambda_1(e^s - 1)).

Valid for all $s \in \mathbb{R}$ because the series $\sum (\lambda_1 e^s)^k/k!$ converges for all $s$ .

Part 2

By independence:

M_{X + Y}(s) = M_X(s)\cdot M_Y(s) = \exp(\lambda_1(e^s - 1))\cdot \exp(\lambda_2(e^s - 1)) = \exp((\lambda_1 + \lambda_2)(e^s - 1)).

Part 3

This is the MGF of a Poisson distribution with parameter

\lambda_1 + \lambda_2

. By the MGF uniqueness theorem,

X + Y \sim \text{Poisson}(\lambda_1 + \lambda_2)

. This is the superposition property of independent Poissons.

Part 4

Total order count per minute: $X + Y \sim \text{Poisson}(20 + 15) = \text{Poisson}(35)$ .

Mean: $\mathbb{E}[X + Y] = 35$ orders/minute.
Variance: $\text{Var}(X + Y) = 35$ orders²/minute² (recall Poisson variance equals the rate).
Standard deviation: $\sqrt{35} \approx 5.92$ orders/minute.

Interpretation. In a minute you expect about 35 orders with a standard deviation of about 6. If you observe a burst of 50+ orders in a single minute, that's about 2.5 standard deviations above the mean — unusual but not shocking. If you see 70+, the Poisson assumption is probably wrong (suggesting self-exciting dynamics / Hawkes processes instead). This kind of quick back-of-envelope is exactly what the Poisson-superposition property enables.

Takeaways

Independent Poissons sum to Poisson. A structural property, not just a coincidence. It's a cornerstone of arrival-process modelling.
The MGF of Poisson is a "double-exponential": $\exp(\lambda(e^s - 1))$ . Memorise the shape; it appears in every Poisson-process calculation.
Superposition extends to $n$ streams. If $X_i \sim \text{Poisson}(\lambda_i)$ independent, then $\sum X_i \sim \text{Poisson}(\sum \lambda_i)$ . Same MGF argument. This is why aggregating arrival streams from different trading venues still yields a Poisson process under the independence assumption.
Poisson mean = variance. A hallmark — departures from this identity (overdispersion or underdispersion) signal non-Poisson dynamics.