# Maximum Likelihood Estimation with 2 parameters for a Poisson distribution

I have two observations from a Poisson distribution. The first one ($N_r$) come with a Poisson distribution with mean $k_1$. For the second one ($N_e$) I know that $N_e - M$ also come from the same Poisson distribution with mean $k_1$, but the parameter $M$ is unknown. My observations are $N_r$ and $N_e$. I want to do the MLE for $M$ and $k_1$.

If I write the equations: $$f(N_r, N_e|k_1, M) = \frac{e^{-k_1}k_1^{N_r}}{N_r!}\frac{e^{-k_1}k_1^{N_e-M}}{(N_e-M)!}=\frac{e^{-2k_1}k_1^{N_r+N_e-M}}{N_r!(N_e-M)!}$$

Taking logarithms: $$ln(f) = -2k_1 + (N_r+N_e-M)ln(k_1)-ln(N_r!(N_e-M)!)$$

I can estimate the mean parameter $k_1$ as $$k_1 = \frac{N_r+N_e-M}{2}$$

But I don't know how to estimate the $M$ parameter as I have to deal with the factorial in the derivative. Can anybody help me?

Thank you very much!!

Edit: You can also convert the factorial expression into an expression that uses gamma functions. gamma(n+1) = n!