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

In summary, the conversation discusses using L to represent the parameter of a Poisson distribution. It mentions the distribution of X and Y, both Poisson variables with different parameters, and the sum of aX and bY with real constants a and b. The topic of finding the maximum likelihood estimator of L is also brought up, along with the need for the pmf of S, which is a Poisson variable with parameter mL + nL. Finally, the conversation mentions that the sum of two Poisson processes is also a Poisson process with a combined parameter.
  • #1
JGalway
6
0
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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Related to How Do You Find the MLE of Lambda in a Sum of Two Poisson Distributions?

1. What is a Linear Combination Poisson?

A Linear Combination Poisson (LCP) is a statistical method used to estimate the parameters of a Poisson distribution. It combines multiple Poisson distributions to better fit the data and produce more accurate results.

2. How is LCP different from traditional Poisson regression?

LCP differs from traditional Poisson regression in that it takes into account both the mean and the variance of the Poisson distribution, while traditional regression only considers the mean. This allows LCP to better account for overdispersion in the data.

3. When is LCP typically used?

LCP is often used in situations where the traditional Poisson model does not fit the data well, such as when there is overdispersion or when the data is zero-inflated. It is also useful when there are multiple sources of variation in the data.

4. What are the advantages of using LCP?

Some advantages of using LCP include better fit to the data, improved prediction accuracy, and the ability to handle overdispersion and zero-inflated data. LCP also allows for the incorporation of multiple sources of variation, making it a more comprehensive model.

5. What are the potential limitations of LCP?

LCP may be more complex and challenging to interpret compared to traditional Poisson models. It also requires a larger sample size to accurately estimate the parameters. Additionally, LCP may not be suitable for all types of data and may not improve upon traditional Poisson regression in some cases.

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