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

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To find the maximum likelihood estimator (MLE) of lambda (L) in a sum of two Poisson distributions, the parameters are defined as X~Poisson(nL) and Y~Poisson(mL), where m and n are natural numbers. The sum S = aX + bY involves real constants a and b. The key point is that the sum of two independent Poisson variables results in another Poisson variable with a parameter equal to the sum of the individual parameters, specifically (m+n)L. The challenge lies in determining the probability mass function (pmf) for S to compute the MLE effectively. Understanding the properties of Poisson distributions is crucial for solving this problem.
JGalway
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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.
 
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