Assume we have count data of the number of plants for $n$ sites. Also, assume that the mean among sites is different, whereby $\mu_1 \neq \mu_2 \neq \mu_3, \cdots,\mu_{n-1} \neq \mu_n$, and each of our $n$ datasets are Poisson distributed.
Let's say we have $3$ sampling locations ($x,y$, and $z$), the joint likelihood function is then $$\begin{align} \ell(\mu) &=\prod_{i=1}^n \frac{e^{-\mu_1} \mu_1^{x_i}}{x_i!} \prod_{i=1}^n \frac{e^{-\mu_2} \mu_2^{y_i}}{y_i!} \prod_{i=1}^n \frac{e^{-\mu_3} \mu_3^{z_i}}{z_i!} \\ &= \prod_{i=1}^n \frac{e^{-\mu_1 -\mu_2 -\mu_3} \mu_1^{x_i} \mu_2^{y_i} \mu_3^{z_i}}{x_i!y_i!z_i!} \end{align}$$
Is this correct?
