Suppose that the joint distribution of the two random variables x and y is $\displaystyle f(x,y)=\frac{\theta e^{-(\beta+\theta)y}(\beta y)^x}{x!}$ where $\displaystyle \beta, \theta>0, y\geq 0, x=1,2,....$

a) Find the maximum likelihood estimators of $\displaystyle \beta$ and $\displaystyle \theta$ and their asymptotic joint distribution.

b) Find the maximum likelihood estimator of $\displaystyle \frac{\theta}{\beta+\theta}$ and its asymptotic distribution.


So far I've got $\displaystyle \hat{\beta}_{ML}=\frac{\bar{x}} {\bar{y}}$ and $\displaystyle \hat{\theta}_{ML}=\frac{1} {\bar{y}}$ in a) and that the ML estimator for $\displaystyle \frac{\theta}{\beta+\theta}=\frac{1}{1+\bar{x}}$ in b)

However, I'm stuck on those asymptotic joint distributions, anyone who can help? Perhaps even step by step?

Regards.