How Do You Find the MLE of Lambda in a Sum of Two Poisson Distributions?

[JGalway]

First of all I will use L to denote lambda the parameter of the distribution.
X~Poission(nL), n$\in\Bbb{N}$,
Y~Poisson(mL),m$\in\Bbb{N}$ with m$\ne$n
S= aX+bY a,b real constants.
Given observations x and y find the maximum likelihood estimator of L.

The problem is I don't know what the pmf is for S which as far as I know you need to get the MLE.
Thanks in advance for any feedback.

 

[HallsofIvy]

If x and y are Poisson variables with parameters mL and nL then their sum is a Poisson variable with parameter mL+ nL= (m+n)L.

statistics - How do I sum two Poisson processes? - Mathematics Stack Exchange