# How to write the log likelihood for the sum of independent poisson events

Say that the per each event of TYPE 1, the average number of occurrences is $$\lambda_1$$. Then the likelihood for the number of occurrences in a single event, $$k_1$$, is $$\lambda_1^k / k_1! * e^{-\lambda_1}$$. Now let's say I also have events of TYPE 2 for which the average number of occurrences is $$\lambda_2$$.

I want to write a likelihood for the total number of occurrences across multiple events of each type (i.e., across $$n_1$$ events of TYPE 1 and $$n_2$$ events of TYPE 2)? The main unknown in this case would be the total number of occurrences, $$k$$. The "occurrences" in these two types of events are the same (e.g., number of traffic stops on weekdays [TYPE 1] and weekends [TYPE 2])

As an example, let's say $$\lambda_1 = 1.2$$ and $$\lambda_2 = 1.8$$ and there are $$n_1 = 10$$ events of TYPE 1 and $$n_2 = 20$$ events of TYPE 2. The likelihood should maximize right around $$k = \lambda_1*n_1 + \lambda_2*n*2 = 48$$.

How would I write out such a likelihood?

• type 1 and type are independent or not? Jul 23, 2019 at 17:58
• ... and are you only interested in the distribution of $n_1+n_2$, or do you want the joint distribution of $(n_1, n_2)$? Jul 23, 2019 at 18:11
• Type 1 and Type 2 events are independent and I want the likelihood for $k_{total}$ across multiple events of each type Jul 23, 2019 at 18:21
• Given $n_1$ and $n_2$ are independent and follow Poisson distribution, then $n_1+n_2 = n$ also follows Poisson distribution with parameter $\lambda_1+\lambda_2$. Then you should be able to write the likelihood for $n$. Jul 23, 2019 at 21:07
• So the following would work? $L = (\lambda_1 + \lambda_2)^k / k! * e^{-(n_1 + n_2) * (\lambda_1 + \lambda_2)}$ Jul 29, 2019 at 15:23

You are interested in the probability distribution associated with the distribution of the sum of $$n_1$$ and $$n_2$$ independent Poisson random variates with means $$\lambda_1$$ and $$\lambda_2$$ respectively. The sum $$k$$ of two (or more) independent Poisson variates is distributed Poisson with mean equal to the sum of the individual means:
$$P(k|n_1, \lambda_1, n_2, \lambda_2) = {(n_1\lambda_1+n_2\lambda_2)^k\text{e}^{-(n_1\lambda_1+n_2\lambda_2)} \over k!}$$
This distribution has mean $$\lambda = n_1\lambda_1+n_2\lambda_2$$.
The mode of a Poisson distribution with non-integer mean is the mean rounded down, $$\lfloor \lambda \rfloor$$. If $$\lambda$$ is an integer, then there are two modes: $$\lambda$$ and $$\lambda-1$$. These would be the values of $$k$$ that maximize the probability distribution.
• The goal was to find the value of $k$ occurrences that would maximize the corresponding likelihood given that we know the values of $n_1$, $n_2$, $\lambda_1$, and $\lambda_2$ , but I don't see that parameter in the equation - am I missing something? Also, just to clarify, I treat $n_1$, $n_2$, $\lambda_1$, and $\lambda_2$ as known parameters in this example, but this will be just one component of a larger joint likelihood that will simultaneously estimate multiple parameters - so I would want to include all parameters in the equation (even if they would otherwise be constant) Aug 5, 2019 at 19:35
• Ah, I think I misread your question slightly... will revise my answer. However, you seem to have confused "probability" with "likelihood"; you are trying to find $k$ that maximizes the probability. Aug 5, 2019 at 19:48