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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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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
up vote 1 down vote accepted

A few hints.

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 $$ -\Phi_X^{\prime\prime}(0)=\int_{-\infty}^\infty t^2\phi(t)\,\mathrm{d}t=\mathrm{E}[X^2] $$

Hint 2: the characteristic function of the sum of two random variables is the product of their characteristic functions.

Hint 3: The Riemann-Lebesgue Lemma

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