# probability question on characteristic function

I got a big problem with my exam practice question on characteristic function. Here are two.

Let $X$, $Y$ be two independent random variables with the following characteristic functions: $$\Phi_X(\theta)=\tfrac{1}{4}e^{i\theta} + \tfrac{3}{4}e^{2i\theta} , \quad \Phi_Y(\theta)=\exp(e^{i\theta}-1).$$

(a) Find $E[X^2]$.
(b) Find $P(X+Y)=2$.
(c) Does $X+Y$ admit a Lebesgue density?

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.

Unfortunately, the lecture notes are quite poor. I am wondering if anyone can give me a link of some good notes on this part (characteristic function and convolution). Thanks and regards.

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Please see here and here for how to format your mathematics expressions with LaTeX, and see here for how to use Markdown formatting. If you need to format more advanced math, there are many excellent LaTeX references on the internet, including StackExchange's own TeX.SE site. If you see a piece of LaTeX you want to know the code for on the site, you can right click on it, go to "Show Math As", then choose "TeX Commands". –  Zev Chonoles Jun 9 '12 at 2:00
For the first question, use the fact that you can differentiate the characteristic function. I guess in question b there is a typo. –  Davide Giraudo Jun 9 '12 at 8:49

Hint 1: Since the characteristic function $$\Phi_X(\theta)=\int_{-\infty}^\infty e^{it\theta}\phi(t)\,\mathrm{d}t$$ If we take the first derivative with respect to $\theta$ and evaluate at $0$, we get $$-i\Phi_X^{\prime}(0)=\int_{-\infty}^\infty t\phi(t)\,\mathrm{d}t=\mathrm{E}[X]$$ If we take the second derivative with respect to $\theta$ and evaluate at $0$, we get $$-\Phi_X^{\prime\prime}(0)=\int_{-\infty}^\infty t^2\phi(t)\,\mathrm{d}t=\mathrm{E}[X^2]$$