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

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SUMMARY

The maximum likelihood estimator (MLE) of lambda (L) in a sum of two Poisson distributions can be derived from the properties of Poisson random variables. Given X ~ Poisson(nL) and Y ~ Poisson(mL), where m and n are natural numbers and m ≠ n, the sum S = aX + bY, with a and b as real constants, requires understanding the probability mass function (pmf) of S. The sum of two independent Poisson variables results in another Poisson variable with parameter (m+n)L, which is crucial for calculating the MLE of L based on observed values x and y.

PREREQUISITES
  • Understanding of Poisson distributions and their properties
  • Familiarity with maximum likelihood estimation (MLE) techniques
  • Knowledge of probability mass functions (pmf)
  • Basic statistics involving random variables
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  • Study the derivation of the probability mass function for a sum of Poisson distributions
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Statisticians, data scientists, and researchers working with Poisson processes or involved in statistical modeling and estimation techniques.

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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