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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.

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marked as duplicate by Did, LVK, Rudy the Reindeer, t.b., sdcvvc Aug 27 '12 at 0:32

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The characteristic function is defined to be $E e^{\theta iX + \eta iY}$. What happens when (say) $\eta=0$? –  guy Jun 9 '12 at 15:27

1 Answer 1

$$ \Phi_{X}(\theta)=\Phi_{X,Y}(\theta,0)\qquad \Phi_{Y}(\theta)=\Phi_{X,Y}(0,\theta)\qquad \Phi_{X+Y}(\theta)=\Phi_{X,Y}(\theta,\theta) $$

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