Part (ii) of this question implies that the denominator of E(x) is (alpha + beta). I think it should just be beta. You have to create Probability Density Functions for X and Y: they are not Poisson because not all integer outcomes are possible. They are Poisson divided by e to the minus lambda times beta (for X); and divided by e to the minus lambda times alpha (for Y). This also has consequences for the variance calculations.