How to find the log-likelihood for this density?

Question

We let $(X,Y)$ be stochastic variables with values in $N × R$ determined by $ X = x $ being subtracted from one Poisson distribution with mean $\lambda$ after which Y is subtracted from a gamma distribution with shape parameter x + 1 and scale parameter β, i.e. the simultaneous distribution of (X, Y) has the density: $$f_{(\lambda,\beta)}(x,y)=\frac{\lambda^{x}}{x!}e^{-\lambda}\frac{y^{x}}{\beta^{x+1}x!}e^{-\frac{y}{\beta}}$$ with respect to m×$\nu$ where m is the count measure on $N_0$ and ν is the standard lebesgue measure on $R^+$. I'm not totally sure how to find the log-likelihood function. I have in the solution that: $$l_{n}(\lambda,\beta)=log \lambda \sum_i X_i-log \beta \sum_i X_i-\sum_I Y_i/\beta-n \lambda-n log \beta$$. But how to find it? I have seen some easy example but not that diffucult. Can anyone help me with a guide and some tricks?

Answer 1

It's very easy. The trick is to eliminate any element not depending on the parameter thus your likelihood is

$$L \propto \lambda ^{\Sigma_i X_i}e^{-n \lambda}\beta^{-\Sigma_i(X_i+1)}e^{-\Sigma_i Y_i/\beta}$$

Now take the log and find the solution

$$l=\log\lambda\Sigma_i X_i-n\lambda-\Sigma_i(X_i+1)\log\beta-\frac{\Sigma_i Y_i}{\beta}$$