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

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

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


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