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probability question on characteristic function
It is a problem in my practice exam.
Defined on some common probability space, two random variables $X$, $Y$ have the following joint characteristic function: $\Phi_{X,Y}(\theta,\eta) = \frac{1}{1+\theta^2} \cdot \exp(-i\eta-\eta^2)$
(a) Find $\Phi_X(\theta)$ and $E[X]$ and $E[X^2]$.
(b) Find $\Phi_{X+Y}(\theta)$ and $\operatorname{Var}(X+Y)$.
(c) Prove or disprove that $X+Y$ is absolutely continuous.
Is there any way to calculate $\Phi_X(\theta)$ and $\Phi_Y(\theta)$ from the given joint characteristic function? I think the remaining parts will be easy once I get these two. Thanks and regards.