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.