$X$ and $Y$ are independent Poisson distributed values, means are $θ$ and $2θ$.
Consider the combined estimator $$\gamma = k_1\cdot X + k_2\cdot Y$$ where $k_1$ and $k_2$ are arbitrary constants.
(a) Find the condition on $k_1$ and $k_2$ such that $\gamma$ is an unbiased estimator.
To solve this I let $\theta = E[k_1X + k_2Y] = k_1E[X] + k_2E[Y]$
$\theta = k_1\theta + 2k_2\theta$, $k_1 + 2k_2 = 1$
However I'm stuck when it comes to the next 2 parts:
(b) For $\gamma$ unbiased, show that the variance of the estimator is minimized by taking $k_1 = 1/3$ and $k_2 = 1/3$.
(c) Given observations x and y find the maximum likelihood estimate of θ. and hence show that $\gamma$ is also the maximum likelihood estimator.
I'm not sure where to even start for part (b) and part (c). Any help would be appreciated!